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The Lp dual Minkowski problem about 0 < p < 1 and q > 0

  • Fangxia Lu EMAIL logo and Zhaonian Pu
Published/Copyright: December 31, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

The ( p , q )-th dual curvature measures and the L p dual Minkowski problem were recently introduced by Lutwak, Yang, and Zhang. In this paper, we give a solution to the existence part of the L p dual Minkowski problem about 0 < p < 1 and q > 0 for arbitrary measures. This fills up previously obtained results.

Keywords: (p, q)-th dual curvature measures; Lp dual Minkowski problem; Minkowski problem; convexity
MSC 2010: 52A20; 52A40

1 Introduction

A compact convex subset of R n with nonempty interior is called a convex body. Let K n denote the set of convex bodies in R n , and K o n denote the set of convex bodies in R n with the origin in their interiors. The unit sphere in R n will be denoted by S n − 1 .

For all x ∈ R n , the support function of K ∈ K n is defined by

h ( K , x ) = h K ( x ) = max { x ⋅ y : y ∈ K } ,

where x ⋅ y denotes the standard inner product of x and y .

For K ∈ K n and v ∈ S n − 1 , the supporting hyperplane H ( K , v ) of K at v is defined by

H ( K , v ) = { x ∈ R n : x ⋅ v = h K ( v ) } .

The radial function, ρ K = ρ ( K , ⋅ ) : R n ⧹ { 0 } → R , of K ∈ K n is defined by

ρ ( K , x ) = max { λ : λ x ∈ K } , x ∈ R n ⧹ { 0 } .

Let K ∈ K o n and η ⊂ S n − 1 be a Borel set. The reverse radial Gauss image of η , α K ∗ ( η ) is given by

α K ∗ ( η ) = { u ∈ S n − 1 : ρ K ( u ) u ∈ H ( K , v ) for some v ∈ η } .

Geometric measures and their associated Minkowski problems in the Brunn-Minkowski theory and its generalization are central to the study of convex geometric analysis. In regard to the dual Brunn-Minkowski theory, Huang et al. in [1] recently studied the q -th dual curvature measures: for K ∈ K o n and q ∈ R , the q -th dual curvature measure, C ˜ q ( K , ⋅ ) , defined for every Borel η ⊂ S n − 1 by

(1.1) C ˜ q ( K , η ) = 1 n ∫ α K ∗ ( η ) ρ K q ( u ) d u

is the Borel measure on S n − 1 . It is worth noting that the q -th dual curvature measure inconceivably connects the well-known cone volume measure ( q = n ) and Aleksandrov’s integral curvature ( q = 0 ). These measures have never been linked before.

Huang et al. [1] asked for necessary and sufficient conditions so that a given measure on the unit sphere is precisely the q -th dual curvature measure of a convex body in R n . This problem is called the dual Minkowski problem. The dual Minkowski problem contains critical problems such as the Aleksandrov problem ( q = 0 ), see, e.g., [2,3,4] and the logarithmic Minkowski problem ( q = n ), see, e.g., [5,6,7, 8,9] as special cases. The problem has been completely solved for q < 0 (see [10]), but critical case for q > 0 is still unsolved, see, e.g., [11,12, 13,14,15].

Very recently, Lutwak et al. in [16] introduced a more general version of the q -th dual curvature measure called the ( p , q )-th dual curvature measure. For K ∈ K o n and p , q ∈ R , the ( p , q )-th dual curvature measure C ˜ p , q ( K , ⋅ ) is defined by

(1.2) d C ˜ p , q ( K , ⋅ ) = h K − p d C ˜ q ( K , ⋅ ) .

It should be noted that this definition is slightly weaker than the one defined in [16]. Lutwak et al. [16] showed that special cases of the ( p , q )-th dual curvature measure are the L p surface area measure ( q = n ), the q -th dual curvature measure ( p = 0 ), and the L p integral curvature ( q = 0 ). Regarding the new ( p , q )-th dual curvature measure, the following L p dual Minkowski problem was posed in [16].

The L p dual Minkowski problem: Given a nonzero finite Borel measure μ on the unit sphere S n − 1 and real numbers p , q , what are the necessary and sufficient conditions so that there is a convex body K ∈ K o n satisfying

C ˜ p , q ( K , ⋅ ) = μ ?

When p = 0 , the L p dual Minkowski problem is just the dual Minkowski problem; when q = 0 , it becomes the L p Aleksandrov problem introduced and studied by Huang et al. [17]; when q = n , it reduces to the L p Minkowski problem, proposed in [18], which has been extensively studied, see, e.g., [19,20, 21,22,23, 24,25,26, 27,28,29, 30,31,32, 33,34,35, 36,37,38, 39,40,41, 42,43].

When p > 0 and q < 0 , a complete characterization to the existence part of the L p dual Minkowski problem is given by Huang and Zhao [44].

Theorem 1.1

[44, Theorem 1.2] Let p > 0 , q < 0 , and μ be a non-zero finite Borel measure on S n − 1 . There is a convex body K ∈ K o n so that μ = C ˜ p , q ( K , ⋅ ) if and only if μ is not contained on arbitrary closed hemisphere.

The Orlicz extension of Theorem 1.1 was partially settled in [45], and later completely solved in [46].

When p , q > 0 and p ≠ q , and the given measure is even, Huang and Zhao [44] also presented a complete solution to the existence part of the L p dual Minkowski problem.

Theorem 1.2

[44, Theorem 1.3] Let p , q > 0 , p ≠ q , and μ be a non-zero even Borel measure on S n − 1 . There is an origin-symmetric convex body K in  R n so that μ = C ˜ p , q ( K , ⋅ ) if and only if μ is not contained in arbitrary great subsphere.

The Orlicz version of Theorem 1.2 was obtained in [46].

When p > 1 , q > 0 , and p > q , a sufficient condition on the existence of solutions to the L p dual Minkowski problem is given by Böröczky and Fodor [47] and obtained the following result.

Theorem 1.3

[47, Theorem 1.2] Let p > 1 , q > 0 , and p > q , and let μ be a finite Borel measure on S n − 1 that is not contained on arbitrary closed hemisphere. Then there is a convex body K ∈ K o n so that μ = C ˜ p , q ( K , ⋅ ) .

The Orlicz case of Theorem 1.3 is given in [46].

As we can see those theorems above, when 0 < p < 1 and q > 0 there is no existence result concerning the L p dual Minkowski problem in the general case (without the condition that the measure is even). The aim of this paper is to supplement the situation, which is motivated by the works of Zhu [41], Jian and Lu [48], Chen et al. [22], and Huang and Zhao [44]. Thus, the following result is obtained.

Theorem 1.4

Let 0 < p < 1 , q > 0 , and q ≠ p . If μ is a finite Borel measure on S n − 1 and is not contained on arbitrary closed hemisphere, then there is a convex body K in R n such that μ = C ˜ p , q ( K , ⋅ ) .

Theorem 1.4 contains as special cases the solution to the existence part of the L p Minkowski problem for 0 < p < 1 (see [22]) and of the discrete L p Minkowski problem for 0 < p < 1 (see [41]).

We remark that when p , q < 0 and the given measure is even, the existence part of the L p dual Minkowski problem was independently solved by Huang and Zhao [44] and Gardner et al. [46]. However, this situation for p < 0 and q > 0 has not yet yielded any results, as far as we know.

In the next section, some preliminaries are given. In Section 3, we consider a minimizing problem and give its corresponding solution. In Section 4, we first discuss the discrete case of Theorem 1.4. Then the proof of Theorem 1.4 is completed by approximation.

2 Preliminaries

In this section, some basic facts about convex bodies are collected. The books of Schneider [49], Gardner [50], and Gruber [51] are excellent references regarding convex bodies.

The work is carried out in R n equipped with the standard Euclidean norm. For any x ∈ R n , its Euclidean norm is denoted by ∣ x ∣ = x ⋅ x . The unit ball will be written by B = { x ∈ R n : ∣ x ∣ ≤ 1 } . The set of continuous functions on S n − 1 is denoted by C ( S n − 1 ) and the set of positive functions in C ( S n − 1 ) is written by C + ( S n − 1 ) .

Let us define the Hausdorff distance of two convex bodies K , L in R n as follows:

δ ( K , L ) = max u ∈ S n − 1 ∣ h K ( u ) − h L ( u ) ∣ .

Assume K i is a sequence of convex bodies in R n . We claim that K i converges to a convex body K 0 ⊂ R n if

δ ( K i , K 0 ) → 0 ,

when i → ∞ .

For g ∈ C + ( S n − 1 ) and a closed subset Ω ⊂ S n − 1 not concentrated on arbitrary closed hemisphere, the Aleksandrov body relevant to ( g , Ω ) , written by [ g ] , is the convex body that is defined as follows:

(2.1) [ g ] = ⋂ u ∈ Ω { ξ ∈ R n : ξ ⋅ u ≤ g ( u ) } .

Clearly, h [ g ] ≤ g and [ h K ] = K if Ω = S n − 1 and K ∈ K o n . In fact, for any v ∈ S n − 1 ,

[ g ] ⊂ { ξ ∈ R n : ξ ⋅ v ≤ g ( v ) } ≕ E ,

which implies

h [ g ] ( v ) ≤ h E ( v ) = g ( v ) .

Thus by the arbitrariness of v ∈ S n − 1 ,

h [ g ] ≤ g .

Moreover, on one hand,

h [ h K ] ≤ h K

for K ∈ K o n . On the other hand, for any u ∈ S n − 1

K ⊂ { ξ ∈ R n : ξ ⋅ u ≤ h K ( u ) } ⊂ ⋂ u ∈ S n − 1 { ξ ∈ R n : ξ ⋅ u ≤ g ( u ) } .

This has

h [ h K ] ≥ h K .

Namely,

h [ h K ] = h K .

It was demonstrated that the ( p , q )-th dual curvature measure is weakly convergent in [16]. Namely, if p , q ∈ R , K i ∈ K o n , and K i → K 0 ∈ K o n , then for every f ∈ C ( S n − 1 ) ,

(2.2) lim i → ∞ ∫ S n − 1 f ( v ) d C ˜ p , q ( K i , v ) = ∫ S n − 1 f ( v ) d C ˜ p , q ( K 0 , v ) .

Moreover, it easily follows from (1.1) and (1.2) that for K ∈ K o n and λ > 0 ,

(2.3) C ˜ p , q ( λ K , ⋅ ) = λ q − p C ˜ p , q ( K , ⋅ ) .

For K ∈ K n , the diameter of K is defined as follows:

D ( K ) = max { ∣ x − y ∣ : x , y ∈ K } .

For q ∈ R and K ∈ K n , the q -th dual volume of K , denoted by V ˜ q ( K ) and see [1], is

(2.4) V ˜ q ( K ) = 1 n ∫ S n − 1 ρ K q ( u ) d u .

Let Ω ⊂ S n − 1 represent a closed subset which is not concentrated on arbitrary closed hemisphere, and let f : Ω → R and h 0 : Ω → ( 0 , ∞ ) be continuous. For t ∈ ( − δ , δ ) with δ > 0 , define continuous function h t : Ω → ( 0 , ∞ ) by

log h t ( v ) = log h 0 ( v ) + t f ( v ) + o ( t , v )

for v ∈ Ω , where o ( t , ⋅ ) : Ω → R is continuous and lim t → 0 o ( t , ⋅ ) / t = 0 .

The next variational formula, see [1, Theorem 4.5], is important in the proof of our main result.

Lemma 2.1

Let [ h t ] be the Aleksandrov body associated with ( h t , Ω ) . Then for q ≠ 0

(2.5) lim t → 0 V ˜ q ( [ h t ] ) − V ˜ q ( [ h 0 ] ) t = q ∫ Ω f ( v ) d C ˜ q ( [ h 0 ] , v ) .

3 The minimization problem

To resolve the Minkowski problem by variational method, the first crucial step is to find an optimization problem whose optimizer is exactly the solution to the Minkowski problem. In this section, we consider a minimization problem and show the existence of a minimizer.

Let g ∈ C + ( S n − 1 ) and μ is a finite discrete measure on S n − 1 which is not contained on arbitrary closed hemisphere of S n − 1 . For the Aleksandrov body relevant to ( g , supp ( μ ) ) , it is denoted by [ g ] μ . Therefore, for 0 < p < 1 , the function, Φ g , μ : [ g ] μ → R , is defined as follows:

(3.1) Φ g , μ ( ξ ) = ∫ S n − 1 ( g ( u ) − ξ ⋅ u ) p d μ ( u ) = ∫ supp ( μ ) ( g ( u ) − ξ ⋅ u ) p d μ ( u ) .

Now, take into account the next minimizing problem:

(3.2) inf { sup ξ ∈ [ g ] μ Φ g , μ ( ξ ) : g ∈ C + ( S n − 1 ) and V ˜ q ( [ g ] μ ) = 1 } .

The solution of problem (3.2) will be given after the following Lemmas 3.1 and 3.2.

Lemma 3.1

Let 0 < p < 1 , if μ is a finite discrete measure on S n − 1 and is not contained on arbitrary closed hemisphere of S n − 1 , then Φ g , μ is strictly concave on [ g ] μ for g ∈ C + ( S n − 1 ) .

Proof

For 0 < p < 1 , t p be strictly concave on [ 0 , + ∞ ) , and for ξ ∈ [ g ] μ and u ∈ supp ( μ ) , we have

g ( u ) − ξ ⋅ u ≥ h [ g ] μ ( u ) − ξ ⋅ u ≥ 0 .

Therefore, for 0 < λ < 1 and ξ 1 , ξ 2 ∈ [ g ] μ ,

Φ g , μ ( λ ξ 1 + ( 1 − λ ) ξ 2 ) = ∫ S n − 1 ( g ( u ) − ( λ ξ 1 + ( 1 − λ ) ξ 2 ) ⋅ u ) p d μ ( u ) = ∫ S n − 1 ( λ ( g ( u ) − ξ 1 ⋅ u ) + ( 1 − λ ) ( g ( u ) − ξ 2 ⋅ u ) ) p d μ ( u ) ≥ λ ∫ S n − 1 ( g ( u ) − ξ 1 ⋅ u ) p d μ ( u ) + ( 1 − λ ) ∫ S n − 1 ( g ( u ) − ξ 2 ⋅ u ) p d μ ( u ) = λ Φ g , μ ( ξ 1 ) + ( 1 − λ ) Φ g , μ ( ξ 2 ) ,

this is also equivalent to proving that

g ( u ) − ξ 1 ⋅ u = g ( u ) − ξ 2 ⋅ u

for arbitrary u ∈ supp ( μ ) , namely,

( ξ 1 − ξ 2 ) ⋅ u = 0 .

Since μ is not contained in arbitrary closed hemisphere, and supp ( μ ) spans the whole space R n . Therefore, we can conclude

ξ 1 = ξ 2 ,

this yields that Φ g , μ is strictly concave on [ g ] μ .□

Lemma 3.2

Let 0 < p < 1 , if μ is a finite discrete measure on S n − 1 and is not contained on arbitrary closed hemisphere of S n − 1 , then for g ∈ C + ( S n − 1 ) , there is a unique ξ g ∈ int ( [ g ] μ ) , which ξ g depends continuously on g so that

Φ g , μ ( ξ g ) = sup ξ ∈ [ g ] μ Φ g , μ ( ξ ) .

Proof

Let Φ g , μ be strictly concave and continuous on [ g ] μ . Thus, there is a unique ξ g ∈ [ g ] μ so that

(3.3) Φ g , μ ( ξ g ) = sup ξ ∈ [ g ] μ Φ g , μ ( ξ ) .

We will show that ξ g ∈ int ( [ g ] μ ) . If not, ξ g is on the boundary, ∂ ( [ g ] μ ) , ξ g ∈ ∂ ( [ g ] μ ) .

Recalling the definition of [ g ] μ ,

[ g ] μ = ⋂ u ∈ supp ( μ ) { ξ ∈ R n : ξ ⋅ u ≤ g ( u ) } .

It is easy to see that there is u ∈ supp ( μ ) so that

(3.4) ξ g ⋅ u = g ( u ) .

Otherwise, for any u ∈ supp ( μ ) , there is ξ g ⋅ u < g ( u ) . Then, for some δ 1 > 0 and any u ∈ supp ( μ ) , we have

ξ g ⋅ u + δ 1 < g ( u ) ,

i.e.,

( ξ g + δ 1 u ) ⋅ u < g ( u ) .

This yields that ξ g ∈ int ( [ g ] μ ) , which is a contradiction.

Let

(3.5) supp ( μ ) = A ∪ B ,

where

A ≔ { u ∈ supp ( μ ) : ξ g ⋅ u = g ( u ) }

and

B ≔ { u ∈ supp ( μ ) : ξ g ⋅ u < g ( u ) } .

Then, from (3.4), and μ is not contained on arbitrary closed hemisphere of S n − 1 , we can observe that A and B are two disjoint nonempty sets. According to the definition of set A , and noting g ∈ C + ( S n − 1 ) , there exists a unit vector u 0 ∈ S n − 1 so that

(3.6) u 0 ⋅ u < 0 ,

for all u ∈ A . On the basis of these facts that B is a closed subset of S n − 1 and ξ g ⋅ u − g ( u ) , u ∈ S n − 1 is continuous, it follows from that there is a positive constant a > 0 so that

(3.7) ξ g ⋅ u + 2 a < g ( u ) ,

for arbitrary u ∈ B . Therefore, for arbitrary 0 < λ < 2 a and arbitrary u ∈ supp ( μ ) , we obtain

( ξ g + λ u 0 ) ⋅ u < g ( u ) .

This means that there is some δ 2 > 0 so that

( ξ g + λ u 0 + δ 2 u ) ⋅ u < g ( u ) ,

for all u ∈ supp ( μ ) , i.e.,

ξ ( λ ) ≔ ξ g + λ u 0 ∈ int ( [ g ] μ ) .

By definitions (3.1) and (3.5), it follows that

(3.8) Φ g , μ ( ξ ( λ ) ) − Φ g , μ ( ξ g ) = ∫ A ∪ B ( g ( u ) − ξ ( λ ) ⋅ u ) p d μ ( u ) − ∫ A ∪ B ( g ( u ) − ξ g ⋅ u ) p d μ ( u ) = ∫ A ( g ( u ) − ξ ( λ ) ⋅ u ) p d μ ( u ) + ∫ B ( g ( u ) − ξ ( λ ) ⋅ u ) p − ( g ( u ) − ξ g ⋅ u ) p d μ ( u ) .

For all u ∈ A and some constant δ 3 > 0 , inequality (3.6) is strengthened as follows:

u 0 ⋅ u < − δ 3 < 0 .

Thus, for all u ∈ A ,

g ( u ) − ξ ( λ ) ⋅ u = − λ u 0 ⋅ u > λ δ 3 .

This has

(3.9) ∫ A ( g ( u ) − ξ ( λ ) ⋅ u ) p d μ ( u ) > ∫ A ( λ δ 3 ) p d μ ( u ) = ( λ δ 3 ) p μ ( A ) .

From (3.7), we get that for arbitrary u ∈ B and 0 < λ < a .

g ( u ) − ξ ( λ ) ⋅ u = g ( u ) − ξ g ⋅ u − λ u 0 ⋅ u > 2 a − λ > a .

For 0 < p < 1 , this has

∣ ( g ( u ) − ξ ( λ ) ⋅ u ) p − ( g ( u ) − ξ g ⋅ u ) p ∣ < p a p − 1 ∣ − λ u 0 ⋅ u ∣ ≤ λ p a p − 1 .

Thus,

(3.10) ∫ B ( g ( u ) − ξ ( λ ) ⋅ u ) p − ( g ( u ) − ξ g ⋅ u ) p d μ ( u ) ≤ ∫ B ( g ( u ) − ξ ( λ ) ⋅ u ) p − ( g ( u ) − ξ g ⋅ u ) p d μ ( u ) ≤ ∫ B ∣ ( g ( u ) − ξ ( λ ) ⋅ u ) p − ( g ( u ) − ξ g ⋅ u ) p ∣ d μ ( u ) < λ p a p − 1 μ ( B ) .

Associated with (3.8), (3.9) and (3.10), we have

Φ g , μ ( ξ ( λ ) ) − Φ g , μ ( ξ g ) > ( λ δ 3 ) p μ ( A ) − λ p a p − 1 μ ( B ) = λ p ( δ 3 p μ ( A ) − p λ 1 − p a p − 1 μ ( B ) ) .

We can choose 0 < λ 0 < a small enough so that ξ ( λ 0 ) ∈ Int ( [ g ] μ ) and

Φ g , μ ( ξ ( λ 0 ) ) > Φ g , μ ( ξ g ) ,

which is a contradiction since the maximum of Φ g , μ is achieved at the point ξ g from (3.3). Therefore, ξ g ∈ int ( [ g ] μ ) .

Let g ∈ C + ( S n − 1 ) , { g k } ⊂ C + ( S n − 1 ) be arbitrary sequence of functions and uniformly converging to g on S n − 1 . We next show that ξ g k converges to ξ g in R n .

Note that the fact, see [49, Lemma 7.5.2], that [ g k ] μ → [ g ] μ as g k → g uniformly on S n − 1 . Since ξ g k ∈ [ g k ] μ , ξ g k is bounded. Thus, we let { ξ g k i } ⊂ { ξ g k } be arbitrary convergent subsequence and ξ g k i → ξ 0 as i → + ∞ . We will prove ξ 0 = ξ g .

Let ξ ∈ [ g ] μ . Thus, [ g k i ] μ → [ g ] μ as i → + ∞ , there is a sequence with ξ k i ∈ [ g k i ] μ so that ξ k i → ξ as i → + ∞ . Then,

Φ g , μ ( ξ ) = ∫ S n − 1 ( g ( u ) − ξ ⋅ u ) p d μ ( u ) = lim i → + ∞ ∫ S n − 1 ( g k i ( u ) − ξ k i ⋅ u ) p d μ ( u ) ≤ lim i → + ∞ ∫ S n − 1 ( g k i ( u ) − ξ g k i ⋅ u ) p d μ ( u ) = ∫ S n − 1 ( g ( u ) − ξ 0 ⋅ u ) p d μ ( u ) = Φ g , μ ( ξ 0 ) .

This has

sup ξ ∈ [ g ] μ Φ g , μ ( ξ ) = Φ g , μ ( ξ 0 ) .

By the uniqueness of ξ g , it follows that ξ g = ξ 0 , which proves ξ g k → ξ g .□

We are now ready to show the solution of problem (3.2).

Theorem 3.3

Let 0 < p < 1 , if μ is a finite discrete measure on S n − 1 and is not contained on arbitrary closed hemisphere of S n − 1 , then there is a function h ∈ C + ( S n − 1 ) with ξ h = o and V ˜ q ( [ h ] μ ) = 1 so that

(3.11) Φ h , μ ( o ) = inf { sup ξ ∈ [ g ] μ Φ g , μ ( ξ ) : g ∈ C + ( S n − 1 ) and V ˜ q ( [ g ] μ ) = 1 } .

Proof

Let { g k } ⊂ C + ( S n − 1 ) , V ˜ q ( [ g k ] μ ) = 1 , and

(3.12) lim k → + ∞ Φ g k , μ ( ξ g k ) = inf { sup ξ ∈ [ g ] μ Φ g , μ ( ξ ) : g ∈ C + ( S n − 1 ) and V ˜ q ( [ g ] μ ) = 1 } .

Define h k = h [ g k ] μ . We can observe that for u ∈ supp ( μ ) ,

h k ( u ) ≤ g k ( u ) ,

where [ h k ] μ = [ h [ g k ] μ ] μ = [ g k ] μ . Note that o ∈ int ( [ g k ] μ ) . Thus, h k ∈ C + ( S n − 1 ) . For any ξ ∈ [ h k ] μ = [ g k ] μ , we have

Φ h k , μ ( ξ ) = ∫ supp ( μ ) ( h k ( u ) − ξ ⋅ u ) p d μ ( u ) ≤ ∫ supp ( μ ) ( g k ( u ) − ξ ⋅ u ) p d μ ( u ) = Φ g k , μ ( ξ ) .

This obtains

sup ξ ∈ [ h k ] μ Φ h k , μ ( ξ ) ≤ sup ξ ∈ [ g k ] μ Φ g k , μ ( ξ ) .

Thus,

(3.13) lim k → + ∞ sup ξ ∈ [ h k ] μ Φ h k , μ ( ξ ) ≤ lim k → + ∞ sup ξ ∈ [ g k ] μ Φ g k , μ ( ξ ) .

Note that h k ∈ C + ( S n − 1 ) and V ˜ q ( [ h k ] μ ) = V ˜ q ( [ g k ] μ ) = 1 . Then,

(3.14) lim k → + ∞ sup ξ ∈ [ h k ] μ Φ h k , μ ( ξ ) ≥ lim k → + ∞ sup ξ ∈ [ g k ] μ Φ g k , μ ( ξ ) .

Combining (3.13) with (3.14), we have

lim k → + ∞ sup ξ ∈ [ h k ] μ Φ h k , μ ( ξ ) = lim k → + ∞ sup ξ ∈ [ g k ] μ Φ g k , μ ( ξ ) .

This, together with (3.12), has

lim k → + ∞ Φ h k , μ ( ξ h k ) = inf { sup ξ ∈ [ g ] μ Φ g , μ ( ξ ) : g ∈ C + ( S n − 1 ) and V ˜ q ( [ g ] μ ) = 1 } .

From Lemma 3.2, we see ξ h k ∈ int ( [ h k ] μ ) and

Φ h k , μ ( ξ h k ) = sup ξ ∈ [ h k ] μ Φ h k , μ ( ξ ) .

Recalling [ h k ] μ = [ g k ] μ , we get h k = h [ g k ] μ = h [ h k ] μ , namely, h k is the support function of [ h k ] μ as well. For x ∈ R n , we calculate

Φ h ( [ h k ] μ + x ) , μ ( ξ h k + x ) = Φ h k , μ ( ξ h k ) .

Therefore, we can find a sequence, again denoted by { h k } ⊂ C + ( S n − 1 ) , V ˜ q ( [ h k ] μ ) = 1 , and ξ h k = o such that

(3.15) lim k → + ∞ Φ h k , μ ( o ) = inf { sup ξ ∈ [ g ] μ Φ g , μ ( ξ ) : g ∈ C + ( S n − 1 ) and V ˜ q ( [ g ] μ ) = 1 } .

That is to say that { h k } is uniformly bounded on S n − 1 . If not, there exists a subsequence of { h k } , also written by { h k } , such that

lim k → + ∞ max u ∈ S n − 1 h k ( u ) = + ∞ .

Let R k = max u ∈ S n − 1 h k ( u ) = h k ( u k ) for u k ∈ S n − 1 . Since { u k } ⊂ S n − 1 , it can be seen from the compactness of S n − 1 that there exists a convergent subsequence, say { u k } , assuming

lim k → + ∞ u k = u 0 ∈ S n − 1 .

Since supp ( μ ) is not contained in arbitrary closed hemisphere, thus, there is some u ′ ∈ supp ( μ ) so that

u ′ ⋅ u 0 > 0 .

Let b = 1 2 ( u ′ ⋅ u 0 ) > 0 . Then there is k 0 ∈ N so that as k ≥ k 0 ,

u ′ ⋅ u k > b .

Note that R k u k ∈ [ h k ] μ . Thus, when k ≥ k 0 ,

h k ( u ′ ) ≥ R k ( u ′ ⋅ u k ) > R k b .

It then follows from that μ be a finite discrete measure, we obtain that for k ≥ k 0 and 0 < p < 1 ,

(3.16) lim k → + ∞ Φ h k , μ ( o ) = lim k → + ∞ ∫ S n − 1 h k p ( u ) d μ ( u ) ≥ lim k → + ∞ h k p ( u ′ ) μ ( u ′ ) > lim k → + ∞ ( R k b ) p μ ( u ′ ) = + ∞ .

Let h ′ ∈ C + ( S n − 1 ) and V ˜ q ( [ h ′ ] μ ) = 1 . Thus,

lim k → + ∞ Φ h k , μ ( o ) ≤ Φ h ′ , μ ( ξ h ′ ) = ∫ S n − 1 ( h ′ ( u ) − ξ h ′ ⋅ u ) p d μ ( u ) < + ∞

this contradicts with (3.16). Consequently, { h k } is uniformly bounded.

According to the Blaschke selection theorem, { h k } has a convergent subsequence, also denoted by { h k } , letting h k → h on S n − 1 as k → + ∞ . Thus, there are h = h [ h ] μ and [ h k ] μ → [ h ] μ . Moreover, we have h ≥ 0 and V ˜ q ( [ h ] μ ) = 1 . From Lemma 3.2, we see

o = lim k → + ∞ ξ h k = ξ h ∈ int ( [ h ] μ ) .

Thus, h > 0 , and associated with (3.15) we get

Φ h , μ ( o ) = inf { sup ξ ∈ [ g ] μ Φ g , μ ( ξ ) : g ∈ C + ( S n − 1 ) and V ˜ q ( [ g ] μ ) = 1 } .

The proof of Theorem 3.3 is completed.□

4 Solving the L p dual Minkowski problem

In the following, we first prove that the solution h which is obtained in Theorem 3.3 is exactly the solution to the discrete case of Theorem 1.4. Then using approximation, Theorem 1.4 is proved.

Theorem 4.1

Let 0 < p < 1 and q ≠ 0 . If μ is a finite discrete measure on S n − 1 and is not contained on arbitrary closed hemisphere of S n − 1 , then there is a function h ∈ C + ( S n − 1 ) satisfying (3.11) and a positive constant c > 0 so that

μ = c C ˜ p , q ( [ h ] μ , ⋅ ) , where c = ∫ S n − 1 h p ( u ) d μ ( u ) .

Proof

By Theorem 3.3, we can see that there is a function h ∈ C + ( S n − 1 ) with ξ h = o and V ˜ q ( [ h ] μ ) = 1 so that

Φ h , μ ( o ) = inf { sup ξ ∈ [ g ] μ Φ g , μ ( ξ ) : g ∈ C + ( S n − 1 ) and V ˜ q ( [ g ] μ ) = 1 } .

For arbitrary f ∈ C ( S n − 1 ) and t ∈ ( − δ , δ ) where δ > 0 is small enough, we have

ϱ t = h e t f .

Then,

log ϱ t = log h + t f .

From Lemma 2.1, we have for q ≠ 0 ,

(4.1) lim t → 0 V ˜ q ( [ ϱ t ] μ ) − V ˜ q ( [ h ] μ ) t = q ∫ supp ( μ ) f ( u ) d C ˜ q ( [ h ] μ , u ) = q ∫ S n − 1 f ( u ) d C ˜ q ( [ h ] μ , u ) .

Let g t = γ ( t ) ϱ t , where

γ ( t ) = V ˜ q ( [ ϱ t ] μ ) − 1 q .

Then g t ∈ C + ( S n − 1 ) and V ˜ q ( [ g t ] μ ) = 1 . Since ϱ 0 = h , it follows from (4.1) that

(4.2) lim t → 0 g t − g 0 t = − h ∫ S n − 1 f ( u ) d C ˜ q ( [ h ] μ , u ) + h f .

Let ξ ( t ) = ξ g t and

(4.3) Φ μ ( t ) = sup ξ ∈ [ g t ] μ ∫ S n − 1 ( g t ( u ) − ξ ⋅ u ) p d μ ( u ) = ∫ S n − 1 ( g t ( u ) − ξ ( t ) ⋅ u ) p d μ ( u ) .

This together with the fact that ξ ( t ) ∈ int ( [ g t ] μ ) has

(4.4) ∫ S n − 1 ( g t ( u ) − ξ ( t ) ⋅ u ) p − 1 u i d μ ( u ) = 0

for i = 1 , … , n , where u = ( u 1 , … , u n ) T . Note that ξ ( 0 ) = ξ h = o . Then taking t = 0 in (4.4), we have

(4.5) ∫ S n − 1 h p − 1 ( u ) u i d μ ( u ) = 0

for i = 1 , … , n . Hence,

(4.6) ∫ S n − 1 h p − 1 ( u ) u d μ ( u ) = 0 .

Let

F i ( t , ξ 1 , … , ξ n ) = ∫ S n − 1 ( g t ( u ) − ( ξ 1 u 1 + ⋯ + ξ n u n ) ) p − 1 u i d μ ( u )

for i = 1 , … , n . Then,

∂ F i ∂ ξ j = ( 1 − p ) ∫ S n − 1 ( g t ( u ) − ( ξ 1 u 1 + ⋯ + ξ n u n ) ) p − 2 u i u j d μ ( u ) .

Let F = ( F 1 , … , F n ) and ξ = ( ξ 1 , … , ξ n ) . Thus,

∂ F ∂ ξ ( 0 , … , 0 ) n × n = ( 1 − p ) ∫ S n − 1 h p − 2 ( u ) u u T d μ ( u ) ,

and u u T is an n × n matrix.

On account of μ is not contained in arbitrary closed hemisphere, and supp ( μ ) spans the whole space R n . Then, for arbitrary x ∈ R n with x ≠ 0 , there is a u i 0 ∈ supp ( μ ) so that u i 0 ⋅ x ≠ 0 . Consequently, for 0 < p < 1 we get

x T ∂ F ∂ ξ ( 0 , … , 0 ) x = x T ( 1 − p ) ∫ S n − 1 h p − 2 ( u ) u u T d μ ( u ) x = ( 1 − p ) ∫ S n − 1 h p − 2 ( u ) ( x ⋅ u ) 2 d μ ( u ) ≥ ( 1 − p ) h p − 2 ( u i 0 ) ( x ⋅ u i 0 ) 2 μ ( u i 0 ) > 0 .

This suggests that ∂ F ∂ ξ ( 0 , … , 0 ) is positive definite, namely,

det ∂ F ∂ ξ ( 0 , … , 0 ) ≠ 0 .

From the implicit function theorem, the facts that F i ( 0 , … , 0 ) = 0 follows by equation (4.5) for i = 1 , … , n , and ∂ F i ∂ ξ j is continuous on a neighborhood of ( 0 , … , 0 ) for all 1 ≤ i , j ≤ n , we conclude that

ξ ′ ( 0 ) = ( ξ 1 ′ ( 0 ) , … , ξ n ′ ( 0 ) )

exists.

Since Φ μ ( 0 ) = Φ h , μ ( o ) and Φ μ ( t ) = Φ g t , μ ( ξ g t ) and note that g t ∈ C + ( S n − 1 ) and V ˜ q ( [ g t ] μ ) = 1 , by Theorem 3.3 we have

Φ μ ( t ) ≥ Φ μ ( 0 ) ,

i.e., Φ μ ( 0 ) is an extremum of Φ μ ( t ) . Therefore, by (4.2) and (4.6) we get

0 = 1 p Φ μ ′ ( 0 ) = ∫ S n − 1 h p − 1 ( u ) − h ( u ) ∫ S n − 1 f ( u ) d C ˜ q ( [ h ] μ , u ) + h ( u ) f ( u ) − ξ ′ ( 0 ) ⋅ u d μ ( u ) = − ∫ S n − 1 h p ( u ) d μ ( u ) ∫ S n − 1 f ( u ) d C ˜ q ( [ h ] μ , u ) + ∫ S n − 1 h p ( u ) f ( u ) d μ ( u ) − ∫ S n − 1 ξ ′ ( 0 ) ⋅ h p − 1 ( u ) u d μ ( u ) = ∫ S n − 1 h p ( u ) f ( u ) d μ ( u ) − c ∫ S n − 1 f ( u ) d C ˜ q ( [ h ] μ , u ) ,

where

c = ∫ S n − 1 h p ( u ) d μ ( u ) > 0 .

That is, for all f ∈ C ( S n − 1 ) ,

∫ S n − 1 h p ( u ) f ( u ) d μ ( u ) = c ∫ S n − 1 f ( u ) d C ˜ q ( [ h ] μ , u ) .

Since h = h [ h ] μ . Then,

d μ ( u ) = c h [ h ] μ − p ( u ) d C ˜ q ( [ h ] μ , u ) .

Associated with (1.2), there is

d μ ( u ) = c d C ˜ p , q ( [ h ] μ , u ) ,

namely,

μ = c C ˜ p , q ( [ h ] μ , ⋅ ) . □

Now, we have prepared enough to finish the proof of Theorem 1.4.

Proof of Theorem 1.4

The following proof is motivated by the work of [49, Theorem 8.2.2], for a given finite Borel measure μ on S n − 1 , which is not contained on arbitrary closed hemisphere, then, we can find a sequence with finite discrete measure { μ j } on S n − 1 so that μ j ( S n − 1 ) = μ ( S n − 1 ) and μ j → μ when j → + ∞ . Especially, μ j is not contained on arbitrary closed hemisphere, on the grounds of Theorem 4.1, for every μ j there is a positive constant c j > 0 and a function h j ∈ C + ( S n − 1 ) so that

(4.7) μ j = c j C ˜ p , q ( [ h j ] μ j , ⋅ ) ,

where

c j = ∫ S n − 1 h j p ( u ) d μ j ( u ) .

Moreover, h j satisfies that ξ h j = o , V ˜ q ( [ h j ] μ j ) = 1 , and

Φ h j , μ j ( o ) = inf { sup ξ ∈ [ g ] μ j Φ g , μ j ( ξ ) : g ∈ C + ( S n − 1 ) and V ˜ q ( [ g ] μ j ) = 1 } ,

where

[ g ] μ j = ⋂ u ∈ supp ( μ j ) { ξ ∈ R n : ξ ⋅ u ≤ g ( u ) }

and

Φ g , μ j ( ξ ) = ∫ S n − 1 ( g ( u ) − ξ ⋅ u ) p d μ j ( u ) = ∫ supp ( μ j ) ( g ( u ) − ξ ⋅ u ) p d μ j ( u ) .

Let m j = Φ h j , μ j ( o ) . Let us prove that m j is uniformly bounded. The Aleksandrov body is related to ( 1 , supp ( μ j ) ), denoted by [ 1 ] μ j . Let

g ^ j = 1 V ˜ q ( [ 1 ] μ j ) 1 q .

Then we see [ g ^ j ] μ j = g ^ j [ 1 ] μ j . Thus, there is V ˜ q ( [ g ^ j ] μ j ) = 1 . Note that μ j ( S n − 1 ) = μ ( S n − 1 ) . Hence,

(4.8) m j = Φ h j , μ j ( o ) ≤ sup ξ ∈ [ g ^ j ] μ j ∫ S n − 1 ( g ^ j ( u ) − ξ ⋅ u ) p d μ j ( u ) ≤ ∫ S n − 1 D ( [ g ^ j ] μ j ) p d μ j ( u ) = D ( [ g ^ j ] μ j ) p μ ( S n − 1 ) = g ^ j p D ( [ 1 ] μ j ) p μ ( S n − 1 ) .

We further show that D ( [ 1 ] μ j ) is uniformly bounded. Otherwise, then there is a sequence of { ξ j } so that ξ j ∈ [ 1 ] μ j and

lim j → + ∞ ∣ ξ j ∣ = + ∞ .

Let ξ ¯ j = ξ j ∣ ξ j ∣ ∈ S n − 1 . By the compactness of S n − 1 , we can assume

lim j → + ∞ ξ ¯ j → ξ ∈ S n − 1 .

On the other hand, supp ( μ ) is not contained on arbitrary closed hemisphere, there is w ∈ supp ( μ ) so that

(4.9) ξ ⋅ w > 0 .

Let U ( w ) be arbitrary neighborhood of w . Then there is

lim inf j → + ∞ μ j ( U ( w ) ) ≥ μ ( U ( w ) ) > 0 .

Now, choose j sufficiently large that satisfy

U ( w ) ⋂ supp ( μ j ) ≠ ∅ ,

which means that we can find a sequence { w j i } so that

w j i ∈ supp ( μ j i ) and lim i → + ∞ w j i = w .

Note that ξ j i ∈ [ 1 ] μ j i . Therefore,

ξ j i ⋅ w j i ≤ h [ 1 ] μ j i ( w j i ) ≤ 1 ,

i.e.,

ξ ¯ j i ⋅ w j i ≤ 1 ∣ ξ j i ∣ .

Taking the limit, it follows that

ξ ⋅ w ≤ 0 .

This contradicts (4.9). Hence, there is a positive constant M > 0 so that

(4.10) D ( [ 1 ] μ j ) ≤ M ,

for all j ∈ N .

By virtue of B ⊂ [ 1 ] μ j for each j ∈ N , we have for q > 0

(4.11) g ^ j ≤ 1 V ˜ q ( B ) 1 q .

Together with (4.8), (4.10), and (4.11), thus it can be seen that for q > 0 and all j ∈ N ,

(4.12) m j ≤ M p V ˜ q ( B ) − p q μ ( S n − 1 ) ,

namely, m j is uniformly bounded.

Now, let us prove that { h j } is uniformly bounded on S n − 1 . If this assertion is not true, there is a subsequence { h j i } ⊂ { h j } so that

lim i → + ∞ max u ∈ S n − 1 h j i ( u ) = + ∞ .

Let R j i = max u ∈ S n − 1 h j i ( u ) = h j i ( u j i ) , and { u j i } ⊂ S n − 1 . Then, by the compactness of S n − 1 , we can assume

lim i → + ∞ u j i = u 0 ∈ S n − 1 .

Since supp ( μ ) is not contained on arbitrary closed hemisphere, there is v 0 ∈ supp ( μ ) so that

v 0 ⋅ u 0 > 0 .

Let U ( v 0 ) is a small neighborhood of v 0 so that for every u ∈ U ( v 0 ) , there is

u ⋅ u 0 > 0 .

Let δ ( u ) = 1 2 ( u ⋅ u 0 ) > 0 for u ∈ U ( v 0 ) , and R j i u j i ∈ [ h j i ] μ j i . Then, we can choose sufficiently large i such that for all u ∈ U ( v 0 ) ,

u ⋅ u j i > δ ( u ) , h j i ( u ) ≥ R j i ( u ⋅ u j i ) > R j i δ ( u ) ,

and

μ j i ( U ( v 0 ) ) ≥ μ ( U ( v 0 ) ) > 0 .

Therefore, for i sufficiently large we have

m j i = ∫ S n − 1 h j i p ( u ) d μ j i ( u ) > R j i p ∫ U ( v 0 ) δ ( u ) p d μ j i ( u ) ≥ R j i p ∫ U ( v 0 ) δ ( u ) p d μ ( u ) ,

which implies that m j i → + ∞ when i → + ∞ . This contradicts (4.12). Hence, { h j } is uniformly bounded on S n − 1 .

According to the Blaschke selection theorem, the sequence { h j } exits a convergent subsequence, again denoted by { h j } , supposing that h j → h on S n − 1 as j → + ∞ . This implies h ≥ 0 , [ h j ] μ j → [ h ] μ as j → + ∞ , and

lim j → + ∞ c j = lim j → + ∞ ∫ S n − 1 h j p ( u ) d μ j ( u ) = ∫ S n − 1 h p ( u ) d μ ( u ) ≕ c 0 ≥ 0 .

From this and (2.2), and taking the limit in (4.7), we see

μ = c 0 C ˜ p , q ( [ h ] μ , ⋅ ) .

On the basis of this, we get c 0 ≠ 0 . Let c 0 = λ q − p with q ≠ p and λ > 0 . Then from (2.3) we have

μ = C ˜ p , q ( λ [ h ] μ , ⋅ ) .

This completes the proof of Theorem 1.4.□

Acknowledgements

The authors express gratitude to the reviewers for fruitful comments and suggestions.

  1. Funding information: This work was supported by the National Natural Science Foundation of China (11461019), the Innovation Foundation of the Higher Education Institutions of Gansu Province, China (2021B-255), and the Innovation Foundation of the Higher Education Institutions of Gansu Province, China (2020B-215).

  2. Author contributions: All authors read and approved the final manuscript.

  3. Conflict of interest: Authors state no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2021-03-13
Revised: 2021-09-24
Accepted: 2021-09-27
Published Online: 2021-12-31

© 2021 Fangxia Lu and Zhaonian Pu, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0118
Keywords for this article
(p, q)-th dual curvature measures; Lp dual Minkowski problem; Minkowski problem; convexity
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  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
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  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
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  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
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  26. Hodge-Deligne polynomials of character varieties of free abelian groups
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  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
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  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
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  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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