The Lp
                Minkowski problem for q-torsional rigidity

Bin Chen; Xia Zhao; Weidong Wang; Peibiao Zhao

doi:10.1515/acv-2022-0041

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The L_p Minkowski problem for q-torsional rigidity

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Published/Copyright: November 24, 2022

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From the journal Advances in Calculus of Variations Volume 17 Issue 3

Abstract

In this paper, we introduce the L p q-torsional measure for p ∈ ℝ and q > 1 by the L p variational formula for the q-torsional rigidity of convex bodies without smoothness conditions. Moreover, we achieve the existence of solutions to the L p Minkowski problem with respect to the q-torsional rigidity for discrete measures and general measures when 0 1 .

Keywords: Minkowski problem; Minkowski inequality

MSC 2010: 52A20; 52A40

1 Introduction

The study of Brunn–Minkowski theory of convex bodies (i.e., a compact, convex set) in Euclidean spaces ℝ n has been an active field in the past century, which is developed from a few basic concepts: support functions, Minkowski combinations, and mixed volumes. As we all know, the Minkowski problem is one of the main cornerstones in Brunn–Minkowski theory of convex bodies.

The classical Minkowski problem is to find a convex body K with the prescribed surface area measure S ⁢ ( K , ⋅ ) which is induced by the volume variational, that is, for each convex body L, there holds

(1.1) d d ⁢ t ⁢ V ⁢ ( K + t ⁢ L ) | t = 0 = ∫ 𝕊 n - 1 h ⁢ ( L , ⋅ ) ⁢ 𝑑 S ⁢ ( K , ⋅ ) ,

where K + t ⁢ L is the Minkowski sum, h ⁢ ( L , ⋅ ) is the support function of L, and 𝕊 n - 1 is the unit sphere.

The solution of the classical Minkowski problem has been solved by famous mathematicians such as Minkowski [31, 32], Alexandrov [1, 2], Fenchel and Jessen [12], Lewy [25] and Nirenberg [33], and others.

As an extension of the classical Minkowski problem, the L p Minkowski problem with respect to the L p surface area measure S p ⁢ ( K , ⋅ ) of a convex body K containing the origin in its interior was proposed and studied in [28]. Here S p ⁢ ( K , ⋅ ) is defined by the volume variational under the Firey’s p-sum ([13])

d d ⁢ t ⁢ V ⁢ ( K + p t ⋅ L ) | t = 0 = 1 p ⁢ ∫ 𝕊 n - 1 h ⁢ ( L , ⋅ ) p ⁢ h ⁢ ( K , ⋅ ) 1 - p ⁢ 𝑑 S ⁢ ( K , ⋅ ) = ^ 1 p ⁢ ∫ 𝕊 n - 1 h ⁢ ( L , ⋅ ) p ⁢ 𝑑 S p ⁢ ( K , ⋅ ) ,

for a compact convex set L containing the origin and p ≥ 1 . Obviously, the case of p = 1 is formula (1.1). Since then, the L p Minkowski problem has become an interest central object in convex geometric analysis and has been widely considered, see, e.g., [6, 5, 8, 21, 29, 30, 27, 35, 37]. Moreover, there are various versions of Minkowski problems related to other functionals in Brunn–Minkowski theory, for instance, the dual Minkowski problem [19], the logarithmic Minkowski problem [4] and the Orlicz–Minkowski problem [15, 17, 26].

From the statements above, one knows that the different geometric functionals will induce some new and different geometric measures. In recent years, some geometric functionals with physical backgrounds have been introduced into the Brunn–Minkowski theory, and related Minkowski-type problems have also been gradually studied, see, e.g., [10, 11, 22, 38]. One of them is the q-torsional rigidity, which is exactly the geometrical functional concerned in the present paper.

For convenience, let 𝒦 o n be the set of convex bodies containing the origin o in their interiors, and let C + 2 be the class of convex bodies of C 2 if its boundary has the positive Gauss curvature.

Now, we recall the concept of the q-torsional rigidity and q-torsional measure. Let Ω be the interior of convex body K in ℝ n and Ω ¯ = K . For q > 1 , the q-torsional rigidity T q ⁢ ( K ) is defined by (see [9])

(1.2) 1 T q ⁢ ( K ) = inf ⁡ { ∫ Ω | ∇ ⁡ ϕ | q ⁢ 𝑑 x [ ∫ Ω | ϕ | ⁢ 𝑑 x ] q : ϕ ∈ W 0 1 , q ⁢ ( Ω ) , ∫ Ω | ϕ | ⁢ 𝑑 x > 0 } .

The functional defined in (1.2) admits a minimizer φ ∈ W 0 1 , q ⁢ ( Ω ) , and c ⁢ φ (for some constant c) is the unique positive solution of the following boundary value problem (see [3] or [18])

(1.3) { △ q ⁢ φ = - 1 in ⁢ Ω , φ = 0 on ⁢ ∂ ⁡ Ω ,

where △ q is the q-Laplace operator.

Following this, Huang, Song and Xu ([20]) defined the q-torsional measure as

μ q tor ⁢ ( Ω , η ) = ∫ g - 1 ⁢ ( η ) | ∇ ⁡ φ | q ⁢ 𝑑 ℋ n - 1 ,

for any Borel set η ⊆ 𝕊 n - 1 . Here g : ∂ ⁡ Ω → 𝕊 n - 1 is the Gauss map, and ℋ n - 1 is the ( n - 1 ) -dimensional Hausdorff measure.

Meanwhile, they replaced the volume functional with the q-torsional rigidity in (1.1), and established the following variational formula in smooth case: Let K , L be two convex bodies of class C + 2 , and q > 1 . Then

(1.4) d d ⁢ t ⁢ T q ⁢ ( K + t ⁢ L ) | t = 0 = ( q - 1 ) ⁢ T q ⁢ ( K ) q - 2 q - 1 ⁢ ∫ 𝕊 n - 1 h ⁢ ( L , ξ ) ⁢ 𝑑 μ q tor ⁢ ( K , ξ ) ,

and the q-torsional rigidity formula

T q ⁢ ( K ) 1 q - 1 = q - 1 q + n ⁢ ( q - 1 ) ⁢ ∫ 𝕊 n - 1 h ⁢ ( K , ξ ) ⁢ 𝑑 μ q tor ⁢ ( K , ξ )

holds. Thus μ q tor ⁢ ( K , ⋅ ) can be regarded as induced by formula (1.4). The q - torsional Minkowski problem was posed: Let μ be a finite Borel measure on 𝕊 n - 1 . Under what necessary and sufficient conditions, does there exist a unique convex K such that μ q tor ⁢ ( K , ⋅ ) = μ ?

Remark 1.1.

Equation (1.4) has not been shown to be valid without smooth condition due to the lack of weak convergence result of q-torsional measure μ q tor ⁢ ( K , ⋅ ) . Secondly, the solution to q-torsional Minkowski problem for general convex body and q > 1 is also blank.

Inspired by the important role of the L p volume variational formula in the Brunn–Minkowski theory, and following the work by Huang, Song and Xu ([20]), we now establish the L p variational formula with respect to the q-torsional rigidity of any convex body without smoothness (for a detailed proof, see Theorem 4.1) below. To implement this variational formula, we first need to deal with the weak convergence of the q-torsional measure (see Theorem 3.3), and generalize formula (1.4) to any convex body without smoothness (see Theorem 3.4).

Theorem 1.2.

Let K ∈ K o n , 1 ≤ p < ∞ and q > 1 . If L is a compact convex set containing the origin, then

d d ⁢ t ⁢ T q ⁢ ( K + p t ⋅ L ) | t = 0 = q - 1 p ⁢ T q ⁢ ( K ) q - 2 q - 1 ⁢ ∫ 𝕊 n - 1 h ⁢ ( L , ξ ) p ⁢ h ⁢ ( K , ξ ) 1 - p ⁢ 𝑑 μ q tor ⁢ ( K , ξ ) .

Similar to the definition of L p surface area measure, we can define the so-called L p q-torsional measure as follows.

Definition 1.3.

For K ∈ 𝒦 o n , p ∈ ℝ and q > 1 , the L p q-torsional measure μ p , q tor ⁢ ( K , ⋅ ) of K is defined by

μ p , q tor ⁢ ( K , η ) = ∫ η h ⁢ ( K , ξ ) 1 - p ⁢ 𝑑 μ q tor ⁢ ( K , ξ ) ,

for each Borel set η ⊆ 𝕊 n - 1 . Obviously, the case of p = 1 is just the q-torsional measure.

Naturally, the L p Minkowski problem of the q-torsional rigidity can be proposed as below.

Problem 1.4 ( L p Minkowski problem of q-torsional rigidity).

Suppose that μ is a finite Borel measure on S n - 1 , p ∈ R and q > 1 . What are the necessary and sufficient conditions on μ such that μ is the L p q-torsional measure μ p , q tor ⁢ ( K , ⋅ ) of a convex body K ∈ K o n ?

For Problem 1.4, Sun, Xu and Zhang [36] only proved the uniqueness of solutions with p ≥ 1 and q > 1 in smooth case, but the existence of solutions was not solved. When p > 1 and q = 2 , Chen and Dai [7] obtained the existence and uniqueness of solutions. For the classical case p = 1 , Colesanti and Fimiani [10] proved the existence and uniqueness of solutions when q = 2 .

The main aim of the present paper is to consider the existence of solutions to Problem 1.4. We first present a solution to Problem 1.4 for discrete measures when 0 1 (see Theorem 5.8) as follows:

Theorem 1.5.

Let μ be a finite positive Borel measure on S n - 1 which is not concentrated on any closed hemisphere. If μ is discrete, 0 1 . Then there exists a convex polytope P containing the origin in its interior such that μ p , q tor ⁢ ( P , ⋅ ) = μ .

Theorem 1.5 will yield a solution to Problem 1.4 for general measures when 0 1 (see Theorem 6.2) as below.

Theorem 1.6.

Let μ be a finite positive Borel measure on S n - 1 which is not concentrated on any closed hemisphere. Suppose 0 1 . Then there exists a convex body K ∈ K o n such that μ p , q tor ⁢ ( K , ⋅ ) = μ .

The organization of this paper is as follows. The background materials and some results are introduced in Section 2. In Section3, we show that formula (1.4) is valid for general convex bodies without smoothness condition. In Section 4, based on the conclusions in Section 3, the L p variational formula for the q-torsional rigidity is established. In Sections 5 and 6, we prove the existence of solutions to the L p Minkowski problem of q-torsional rigidity for discrete measures and general measures when 0 1 .

2 Preliminaries

In this section, we introduce some necessary facts about convex bodies that readers can refer to the celebrated books by Gardner [14] and Schneider [34].

2.1 Basic facts on convex body

For K ∈ 𝒦 o n in ℝ n , the support function h ⁢ ( K , ⋅ ) : 𝕊 n - 1 → ℝ of K is defined by

h ⁢ ( K , ζ ) = max ⁡ { ζ ⋅ Y : Y ∈ K } , ζ ∈ 𝕊 n - 1 ,

where “ ⋅ ” stands for the standard inner product in ℝ n . We clearly know that the support function is a homogeneous convex function with degree 1. Let K and L be two convex bodies, the Minkowski combination of K and L is defined by

K + t ⁢ L = { x + t ⁢ y : x ∈ K , y ∈ L } ,

for t > 0 . The K + t ⁢ L is a convex body whose support function is given by

h ⁢ ( K + t ⁢ L , ⋅ ) = h ⁢ ( K , ⋅ ) + t ⁢ h ⁢ ( L , ⋅ ) .

For K , L ∈ 𝒦 o n and p ≥ 1 , the Firey’s p-sum, K + p t ⋅ L , of K and L is defined through its support function

h ⁢ ( K + p t ⋅ L , ⋅ ) p = h ⁢ ( K , ⋅ ) p + t ⁢ h ⁢ ( L , ⋅ ) p .

In particular, when p = 1 , the Firey’s p-sum is just the classical Minkowski sum.

The radial function ρ K : 𝕊 n - 1 → ( 0 , ∞ ) of K is defined by

ρ ⁢ ( K , υ ) = max ⁡ { c > 0 : c ⁢ υ ∈ K } , υ ∈ 𝕊 n - 1 .

The radial map r K : 𝕊 n - 1 → ∂ ⁡ K is

r K ⁢ ( υ ) = ρ K ⁢ ( υ ) ⁢ υ ,

for υ ∈ 𝕊 n - 1 , i.e. r K ⁢ ( υ ) is the unique on ∂ ⁡ K located on the ray parallel to υ and emanating from the origin.

Two convex bodies K , L ∈ 𝒦 o n are said to be homothetic if K = λ ⁢ L + x for some constant λ > 0 and x ∈ ℝ n , particularly, K and L are said to be dilates of each other if K = λ ⁢ L . The Hausdorff metric on 𝒦 o n , d ℋ ⁢ ( ⋅ , ⋅ ) , is used to measure the distance between two convex bodies K , L ∈ 𝒦 o n , and is defined by

d ℋ ⁢ ( K , L ) = max ζ ∈ 𝕊 n - 1 ⁡ | h ⁢ ( K , ζ ) - h ⁢ ( L , ζ ) | = ∥ h ⁢ ( K , ⋅ ) - h ⁢ ( L , ⋅ ) ∥ ∞ .

Denote by C ⁢ ( 𝕊 n - 1 ) the set of continuous functions defined on 𝕊 n - 1 , which is equipped with the metric induced by the maximal norm. Write C + ⁢ ( 𝕊 n - 1 ) for the set of strictly positive functions in C ⁢ ( 𝕊 n - 1 ) .

For a convex body K and ζ ∈ 𝕊 n - 1 , the support hyperplane H ⁢ ( K , ζ ) is defined by

H ⁢ ( K , ζ ) = { Y ∈ ℝ n : ζ ⋅ Y = h ⁢ ( K , ζ ) } .

The half-space H - ⁢ ( K , ζ ) in the direction ζ is defined by

H - ⁢ ( K , ζ ) = { Y ∈ ℝ n : ζ ⋅ Y ≤ h ⁢ ( K , ζ ) } .

The support set ℱ ⁢ ( K , ζ ) in the direction ζ is defined by

ℱ ⁢ ( K , ζ ) = K ∩ H ⁢ ( K , ζ ) .

For a compact set K ∈ ℝ n , the diameter of K is defined by

d ⁢ ( K ) = max ⁡ { | X - Y | : X , Y ∈ K } .

2.2 Aleksandrov body

Given a function f ∈ C + ⁢ ( 𝕊 n - 1 ) , let K ⊂ ℝ n be such

K ¯ := ⋂ ξ ∈ 𝕊 n - 1 { X ∈ ℝ n : X ⋅ ξ ≤ f ⁢ ( ξ ) } .

Since f is both positive and continuous, it follows that K must be a convex body in ℝ n that contains the origin. The K is often called the Aleksandrov body associated with f. For Aleksandrov body K associated with h ⁢ ( K , ⋅ ) , we see that

h ⁢ ( K , ⋅ ) ≤ f .

Let

ω h = { ξ ∈ 𝕊 n - 1 : h ⁢ ( K , ξ ) < f ⁢ ( ξ ) } .

A basic fact established by Aleksandrov is that

S ⁢ ( K , ω h ) = 0 ,

where S ⁢ ( K , ⋅ ) is the surface area measure on ∂ ⁡ K defined by the ( n - 1 ) -Hausdorff measure. Consequently,

h ⁢ ( K , ⋅ ) = f a.e. with respect to ⁢ S ⁢ ( K , ⋅ ) .

Obviously, if f is the support function of K ∈ 𝒦 o n , then K itself is the Aleksandrov body associated with f.

Aleksandrov’s Convergence Lemma.

Suppose that the functions h i ∈ C + ⁢ ( S n - 1 ) have associated Aleksandrov bodies K i ∈ K o n . Then

h i → h ∈ C + ⁢ ( 𝕊 n - 1 ) ⁢ uniformly ⟹ K i → K ⁢ in the Hausdorff metric ,

where K is the Aleksandrov body associated with h.

3 Variational formula for the q-torsional rigidity of general convex bodies

In this section, we prove that formula (1.4) holds for general convex bodies without smoothness conditions. Before this, we first need to obtain the weak convergence of the q-torsional measure.

Let Ω be an open convex bounded set, let Ω i ( i ∈ ℕ ) be a sequence of open convex bounded sets in ℝ n and let φ i , φ ∈ W 0 1 , q ⁢ ( Ω ) . Let K = Ω ¯ and K i = Ω ¯ i , and let, for ℋ n - 1 -a.e. ξ ∈ 𝕊 n - 1

ℏ ⁢ ( ξ ) = | ∇ ⁡ φ ⁢ ( r K ⁢ ( ξ ) ) | ⁢ ( 𝒥 ⁢ ( ξ ) ) 1 q , ℏ i ⁢ ( ξ ) = | ∇ ⁡ φ i ⁢ ( r K i ⁢ ( ξ ) ) | ⁢ ( 𝒥 i ⁢ ( ξ ) ) 1 q ,

where 𝒥 , 𝒥 i are the Jacobian functions introduced in following lemma.

Lemma 3.1 ([22]).

Let Ω be an open convex bounded set that contains the origin o, let K = Ω ¯ and let f : ∂ ⁡ K → R be H n - 1 -integrable. Then

∫ ∂ ⁡ K f ⁢ ( x ) ⁢ 𝑑 ℋ n - 1 ⁢ ( x ) = ∫ 𝕊 n - 1 f ⁢ ( r K ⁢ ( ξ ) ) ⁢ 𝒥 ⁢ ( ξ ) ⁢ 𝑑 ξ ,

where J is defined H n - 1 -a.e. on S n - 1 by

𝒥 ⁢ ( ξ ) = ( ρ K ⁢ ( ξ ) ) n h K ⁢ ( g K ⁢ ( r K ⁢ ( ξ ) ) ) .

Moreover, there exist constants c 1 , c 2 > 0 such that c 1 < J ⁢ ( ξ ) < c 2 for H n - 1 -a.e. ξ ∈ S n - 1 . Furthermore, assume that { K i } i ∈ N is a sequence of bounded convex bodies converging to K with respect to the Hausdorff metric. Define J i : S n - 1 → ( 0 , ∞ ) by

𝒥 i ⁢ ( ξ ) = ( ρ K i ⁢ ( ξ ) ) n h K i ⁢ ( g K i ⁢ ( r K i ⁢ ( ξ ) ) ) , i ∈ ℕ .

Then there exists i 0 ≥ 1 such that if i ≥ i 0 , then J i ⁢ ( ξ ) is bounded from below and above, uniformly with respect to ξ and i, and { J i } converge to J , H n - 1 -a.e. on S n - 1 .

To prove the weak convergence of the q-torsional measure, we need the following lemma.

Lemma 3.2.

Suppose q > 1 . Then there holds

lim i → ∞ ⁡ ∫ 𝕊 n - 1 | ℏ i q ⁢ ( ξ ) - ℏ q ⁢ ( ξ ) | ⁢ 𝑑 ξ = 0 .

For functions φ i , φ ∈ W 0 1 , q ⁢ ( Ω ) . Let φ , φ i be replaced by the q-equilibrium potential in definitions of ℏ and ℏ i , Lemma 3.2 was proved in [11]. The proof of this lemma is very similar and thus omitted here.

Using the above results, we now establish the weak convergence of the q-torsional measure.

Theorem 3.3.

Let Ω be an open convex bounded set and let Ω i ( i ∈ N ) be a sequence of open convex bounded sets in R n . If q > 1 , and Ω ¯ i converges to Ω ¯ in the Hausdorff metric as i → 0 . Then the sequence of measures μ q tor ⁢ ( Ω i , ⋅ ) weakly converges to μ q tor ⁢ ( Ω , ⋅ ) .

Proof.

Let ρ i , ρ and g i , g be the radial functions and Gauss maps of Ω ¯ i , Ω ¯ , respectively. In order to prove that μ q tor ⁢ ( Ω i , ⋅ ) weakly converges to μ q tor ⁢ ( Ω , ⋅ ) , it is sufficient to show that for any continuous function f on 𝕊 n - 1 there holds

(3.1) lim i → ∞ ⁡ ∫ 𝕊 n - 1 f ⁢ ( ξ ) ⁢ 𝑑 μ q tor ⁢ ( Ω i , ξ ) = ∫ 𝕊 n - 1 f ⁢ ( ξ ) ⁢ 𝑑 μ q tor ⁢ ( Ω , ξ ) .

From the definitions of μ q tor ⁢ ( Ω , ⋅ ) and ℏ ⁢ ( ξ ) , and Lemma 3.1, equation (3.1) is equivalent to

(3.2) lim i → ∞ ⁡ ∫ 𝕊 n - 1 f ⁢ ( g i ⁢ ( ρ i ⁢ ( ξ ) ⁢ ξ ) ) ⁢ ℏ i q ⁢ ( ξ ) ⁢ 𝑑 ξ = ∫ 𝕊 n - 1 f ⁢ ( g ⁢ ( ρ ⁢ ( ξ ) ⁢ ξ ) ) ⁢ ℏ q ⁢ ( ξ ) ⁢ 𝑑 ξ .

Note that

| ∫ 𝕊 n - 1 f ⁢ ( g i ⁢ ( ρ i ⁢ ( ξ ) ⁢ ξ ) ) ⁢ ℏ i q ⁢ ( ξ ) ⁢ 𝑑 ξ - ∫ 𝕊 n - 1 f ⁢ ( g ⁢ ( ρ ⁢ ( ξ ) ⁢ ξ ) ) ⁢ ℏ q ⁢ ( ξ ) ⁢ 𝑑 ξ | ≤ | ∫ 𝕊 n - 1 f ⁢ ( g i ⁢ ( ρ i ⁢ ( ξ ) ⁢ ξ ) ) ⁢ [ ℏ i q ⁢ ( ξ ) - ℏ q ⁢ ( ξ ) ] ⁢ 𝑑 ξ |

+ | ∫ 𝕊 n - 1 [ f ( g i ( ρ i ( ξ ) ξ ) ) - f ( g ( ρ ( ξ ) ξ ) ) ] ℏ q ( ξ ) ) d ξ | .

Since f is continuous on 𝕊 n - 1 , by Lemma 3.2, we get

lim i → ∞ ⁡ | ∫ 𝕊 n - 1 f ⁢ ( g i ⁢ ( ρ i ⁢ ( ξ ) ⁢ ξ ) ) ⁢ [ ℏ i q ⁢ ( ξ ) - ℏ q ⁢ ( ξ ) ] ⁢ 𝑑 ξ | = 0 .

Notice that g i converges to g almost everywhere on 𝕊 n - 1 (see [10, Remark 3.5]), and ρ i converges to ρ uniformly, one has g i ⁢ ( ρ i ⁢ ( ξ ) ⁢ ξ ) converges to g ⁢ ( ρ ⁢ ( ξ ) ⁢ ξ ) almost everywhere on 𝕊 n - 1 as i → ∞ . Thus we have

lim i → ∞ | ∫ 𝕊 n - 1 [ f ( g i ( ρ i ( ξ ) ξ ) ) - f ( g ( ρ ( ξ ) ξ ) ) ] ℏ q ( ξ ) ) d ξ | = 0 .

Hence, we obtain (3.2). This completes the proof of Theorem 3.3. ∎

Based on the weak convergence of the q-torsional measure, we can establish variational formula for the q-torsional rigidity of general convex bodies.

For h ∈ C + ⁢ ( 𝕊 n - 1 ) , denote by T q ⁢ ( h ) the q-torsional rigidity of a Aleksandrov body associated with h. Since the Aleksandrov body associated with the support function h K of a K ∈ 𝒦 o n is exactly the K itself, we have

T q ⁢ ( h K ) = T q ⁢ ( K ) .

Let I ⊂ ℝ be an interval containing 0 and denote by

h t ⁢ ( ξ ) = h ⁢ ( t , ξ ) : I × 𝕊 n - 1 → ( 0 , ∞ ) .

For any fixed t ∈ I , let K t ⊂ ℝ n and denote by

K ¯ t = ⋂ ξ ∈ 𝕊 n - 1 { x ∈ ℝ n : x ⋅ ξ ≤ h ⁢ ( t , ξ ) } .

This is the Aleksandrov body associated with h t . The family of convex domains { K t } t ∈ I will be called the family of Aleksandrov bodies with h t .

Theorem 3.4.

Let I ⊂ R be an interval containing 0 in its interior and let

h ⁢ ( t , ξ ) : I × 𝕊 n - 1 → ( 0 , ∞ )

be continuous and such that the convergence in

h ′ ⁢ ( 0 , ξ ) = lim t → 0 ⁡ h ⁢ ( t , ξ ) - h ⁢ ( 0 , ξ ) t

is uniform on S n - 1 . If { K t } t ∈ I is the family of Aleksandrov bodies associated with h t , and q > 1 , then

(3.3) d d ⁢ t ⁢ T q ⁢ ( K t ) | t = 0 = ( q - 1 ) ⁢ T q ⁢ ( K 0 ) q - 2 q - 1 ⁢ ∫ 𝕊 n - 1 h ′ ⁢ ( 0 , ξ ) ⁢ 𝑑 μ q tor ⁢ ( K 0 , ξ ) .

The proof of this theorem regarding of the variation of the q-torsional rigidity is similar to that of their analogues with respect to the volume or the q-capacity (see [34, Lemma 6.5.3] or [11, Lemma 5.1]). Therefore, the proof will be omitted.

4 The L p q-torsional measure

In this section, we define the L p q-torsional measure by L p variational formula for the q-torsional rigidity of general convex bodies.

Theorem 4.1.

Let K ∈ K o n , 1 ≤ p < ∞ and q > 1 . If L is a compact convex set containing the origin, then

d d ⁢ t ⁢ T q ⁢ ( K + p t ⋅ L ) | t = 0 = q - 1 p ⁢ T q ⁢ ( K ) q - 2 q - 1 ⁢ ∫ 𝕊 n - 1 h ⁢ ( L , ξ ) p ⁢ h ⁢ ( K , ξ ) 1 - p ⁢ 𝑑 μ q tor ⁢ ( K , ξ ) .

Proof.

For K , L ∈ 𝒦 o n and 1 ≤ p < ∞ . By the definition of Firey’s p-sum, we have

lim t → 0 ⁡ h ⁢ ( K + p t ⋅ L , ⋅ ) - h ⁢ ( K , ⋅ ) t = lim t → 0 ⁡ ( h ⁢ ( K , ⋅ ) p + t ⁢ h ⁢ ( L , ⋅ ) p ) 1 p - h ⁢ ( K , ⋅ ) t

= 1 p ⁢ ( h ⁢ ( K , ⋅ ) p + t ⁢ h ⁢ ( L , ⋅ ) p ) 1 p - 1 | t = 0 ⁢ h ⁢ ( L , ⋅ ) p

= h ⁢ ( L , ⋅ ) p ⁢ h ⁢ ( K , ⋅ ) 1 - p p .

Combining Theorem 3.4, we have

d d ⁢ t ⁢ T q ⁢ ( K + p t ⋅ L ) | t = 0 = q - 1 p ⁢ T q ⁢ ( K ) q - 2 q - 1 ⁢ ∫ 𝕊 n - 1 h ⁢ ( L , ξ ) p ⁢ h ⁢ ( K , ξ ) 1 - p ⁢ 𝑑 μ q tor ⁢ ( K , ξ ) .

This completes the proof of Theorem 4.1. ∎

We now give the following definition for the new geometric measure produced by the variational formula in Theorem 4.1.

Definition 4.2.

Suppose p ∈ ℝ and q > 1 . For K ∈ 𝒦 o n , the finite Borel measure μ p , q tor ⁢ ( K , ⋅ ) defined, for each Borel set η ⊆ 𝕊 n - 1 , by

μ p , q tor ⁢ ( K , η ) = ∫ η h ⁢ ( K , ⋅ ) 1 - p ⁢ 𝑑 μ q tor ⁢ ( K , ⋅ ) ,

is called the L p q-torsional measure.

Definition 4.3.

Suppose p ∈ ℝ and q > 1 . For K , L ∈ 𝒦 o n , if p ≠ 0 , define

T p , q ⁢ ( K , L ) = q - 1 q + n ⁢ ( q - 1 ) ⁢ T q ⁢ ( K ) q - 2 q - 1 ⁢ ∫ 𝕊 n - 1 h ⁢ ( L , ξ ) p ⁢ 𝑑 μ p , q tor ⁢ ( K , ξ ) ,

and call it the L p mixed q-torsional rigidity of ( K , L ) . Obviously, T 1 , q ⁢ ( K , L ) = T q ⁢ ( K , L ) and T p , q ⁢ ( K , K ) = T q ⁢ ( K ) .

From Definition 4.2, together with the positive homogeneity and weak convergence of μ q tor ⁢ ( K , ⋅ ) , it is not hard to show that there holds the following lemma.

Lemma 4.4.

For K , K i ∈ K o n and i ∈ N , the following hold:

Let p ∈ ℝ and q > 1 , then μ p , q tor ⁢ ( s ⁢ K , ⋅ ) = s n ⁢ ( q - 1 ) + q - p ⁢ μ p , q tor ⁢ ( K , ⋅ ) for s > 0 .
Let p ∈ ℝ and q > 1 . If K i → K in the Hausdorff metric, then μ p , q tor ⁢ ( K i , ⋅ ) → μ p , q tor ⁢ ( K , ⋅ ) weakly, as i → ∞ .

Next, we establish the natural L p extension of the Brunn–Minkowski- and Minkowski-type inequalities for the q-torsional rigidity.

Theorem 4.5.

For K , L ∈ K o n . If 1 1 , we have

T q ⁢ ( K + p L ) p n ⁢ ( q - 1 ) + q ≥ T q ⁢ ( K ) p n ⁢ ( q - 1 ) + q + T q ⁢ ( L ) p n ⁢ ( q - 1 ) + q ,

with equality if and only if K and L are dilates.

Theorem 4.6.

For K , L ∈ K o n . If 1 1 , we have

T p , q ⁢ ( K , L ) n ⁢ ( q - 1 ) + q ≥ T q ⁢ ( K ) n ⁢ ( q - 1 ) + q - p ⁢ T q ⁢ ( L ) p ,

with equality if and only if K and L are dilates.

For p > 1 , q > 1 , and K , L ∈ 𝒦 o n of class C + 2 , Theorem 4.5 and Theorem 4.6 were proved in [36]. When K , L are any convex bodies in 𝒦 o n , the proof of the Theorems 4.5 and 4.6 are very similar and thus omitted.

5 The L p Minkowski problem of the q-torsional rigidity for discrete measures

In this section, we prove the existence of solutions to the L p Minkowski problem of q-torsional rigidity for discrete measure when 0 1 . First, we study an extremal problem under translation transforms. Next, we establish the relationship between extremal problem and L p Minkowski problem of q-torsional rigidity. Finally, we give the solution for discrete measure.

Let 𝒫 be the set of polytopes in ℝ n . Suppose the unit vectors ξ 1 , … , ξ N ( N ≥ n + 1 ) are not concentrated on any closed hemisphere of 𝕊 n - 1 . Let 𝒫 ⁢ ( ξ 1 , … , ξ N ) be the set with P ∈ 𝒫 ⁢ ( ξ 1 , … , ξ N ) such that for fixed a 1 , … , a N ≥ 0 ,

P = ⋂ k = 1 N { x ∈ ℝ n : x ⋅ ξ k ≤ a k } .

Obviously, for P ∈ 𝒫 ⁢ ( ξ 1 , … , ξ N ) , P has at most N facets, and the outer unit normals of P are a subset of { ξ 1 , … , ξ N } . Let 𝒫 N ⁢ ( ξ 1 , … , ξ N ) be the subset of 𝒫 ⁢ ( ξ 1 , … , ξ N ) such that a polytope P ∈ 𝒫 N ⁢ ( ξ 1 , … , ξ N ) if P ∈ 𝒫 ⁢ ( ξ 1 , … , ξ N ) and P has exactly N facets.

Suppose c 1 , … , c N are positive real numbers and the unit vectors ξ 1 , … , ξ N are not concentrated on any closed hemisphere of 𝕊 n - 1 . Let

μ = ∑ k = 1 N c k ⁢ δ ξ k ⁢ ( ⋅ )

be the discrete measure on 𝕊 n - 1 , where δ is the Kronecker delta. For more knowledge about polytopes and discrete measures, one can refer to [16] for details.

5.1 An extremal problem

For Ω ∈ 𝒫 ⁢ ( ξ 1 , … , ξ N ) and 0 < p < 1 , we define the functional Φ ⁢ ( Ω , ⋅ ) : Ω → ℝ as

(5.1) Φ ⁢ ( Ω , x ) = ∑ k = 1 N c k ⁢ ( h ⁢ ( Ω , ξ k ) - x ⋅ ξ k ) p .

We will show that there is a unique point x Ω ∈ Int ⁡ ( Ω ) such that Φ ⁢ ( Ω , x ) attains the maximum.

Lemma 5.1.

Suppose 0 < p < 1 and the unit vectors ξ 1 , … , ξ N are not concentrated on any closed hemisphere of S n - 1 and Ω ∈ P ⁢ ( ξ 1 , … , ξ N ) . Then there exists a unique point x Ω ∈ Int ⁡ ( Ω ) such that

Φ ⁢ ( Ω , x Ω ) = max x ∈ Ω ⁡ Φ ⁢ ( Ω , x ) .

Proof.

Firstly, we prove the uniqueness of the maximal point. Assume x 1 , x 2 ∈ Int ⁡ ( Ω ) and

Φ ⁢ ( Ω , x 1 ) = Φ ⁢ ( Ω , x 2 ) = max x ∈ Ω ⁡ Φ ⁢ ( Ω , x ) .

From (5.1), and using the Jensen inequality, we get

Φ ⁢ ( Ω , 1 2 ⁢ ( x 1 + x 2 ) ) = ∑ k = 1 N c k ⁢ ( h ⁢ ( Ω , ξ k ) - 1 2 ⁢ ( x 1 + x 2 ) ⋅ ξ k ) p

= ∑ k = 1 N c k ⁢ ( 1 2 ⁢ ( h ⁢ ( Ω , ξ k ) - x 1 ⋅ ξ k ) + 1 2 ⁢ ( h ⁢ ( Ω , ξ k ) - x 2 ⋅ ξ k ) ) p

≥ 1 2 ⁢ ∑ k = 1 N c k ⁢ ( h ⁢ ( Ω , ξ k ) - x 1 ⋅ ξ k ) p + 1 2 ⁢ ∑ k = 1 N c k ⁢ ( h ⁢ ( Ω , ξ k ) - x 2 ⋅ ξ k ) p

= 1 2 ⁢ Φ ⁢ ( Ω , x 1 ) + 1 2 ⁢ Φ ⁢ ( Ω , x 2 )

= max x ∈ Ω ⁡ Φ ⁢ ( Ω , x ) .

Since Ω is convex, we have 1 2 ⁢ ( x 1 + x 2 ) ∈ Ω and then the above equality holds. The equality condition of the Jensen inequality means that

h ⁢ ( Ω , ξ k ) - x 1 ⋅ ξ k = h ⁢ ( Ω , ξ k ) - x 2 ⋅ ξ k , k = 1 , … , N ,

that is,

x 1 ⋅ ξ k = x 2 ⋅ ξ k , k = 1 , … , N .

Since the unit vector ξ 1 , … , ξ N are not concentrated on any closed hemisphere, it follows that x 1 = x 2 . Thus the uniqueness is proved.

Next, we prove the existence of the maximal point. Since Φ ⁢ ( Ω , x ) is continuous in x ∈ Ω and Ω is compact, it follows that Φ ⁢ ( Ω , x ) attains its maximum at a point of Ω, denoted by x Ω . Thus we only need to prove x Ω ∈ Int ⁡ ( Ω ) . We use proof by contradiction. Suppose x Ω ∈ ∂ ⁡ Ω with

h ⁢ ( Ω , ξ k ) - x Ω ⋅ ξ k = 0 ,

for k = { i 1 , … , i m } , and

h ⁢ ( Ω , ξ k ) - x Ω ⋅ ξ k > 0 ,

for k = { 1 , … , N } ∖ { i 1 , … , i m } , where 1 ≤ i 1 < ⋯ < i m ≤ N and 1 ≤ m ≤ N - 1 . Fix y 0 ∈ Int ⁡ ( Ω ) and let

ξ 0 = y 0 - x Ω | y 0 - x Ω | .

Then for sufficiently small ε > 0 , it follows that x Ω + ε ⁢ ξ 0 ∈ Int ⁡ ( Ω ) . In the following, we aim to show that Φ ⁢ ( Ω , x Ω + ε ⁢ ξ 0 ) - Φ ⁢ ( Ω , x Ω ) > 0 , which will contradict the maximality of Φ at x Ω . Consequently, x Ω ∈ Int ⁡ ( Ω ) .

Let

(5.2) [ h ⁢ ( Ω , ξ k ) - ( x Ω + ε ⁢ ξ 0 ) ⋅ ξ k ] - [ h ⁢ ( Ω , ξ k ) - x Ω ⋅ ξ k ] = - ( ξ 0 ⋅ ξ k ) ⁢ ε = α k ⁢ ε ,

where α k = - ( ξ 0 ⋅ ξ k ) . Since h ⁢ ( Ω , ξ k ) - x Ω ⋅ ξ k = 0 for k ∈ { i 1 , … , i m } and y 0 is an interior point of Ω, it follows that α k > 0 for k ∈ { i 1 , … , i m } . Let

(5.3) α 0 = min ⁡ { h ⁢ ( Ω , ξ k ) - x Ω ⋅ ξ k : k = { 1 , … , N } ∖ { i 1 , … , i m } } > 0 ,

and choose ε > 0 small enough such that x Ω + ε ⁢ ξ 0 ∈ Int ⁡ ( Ω ) and

(5.4) min ⁡ { h ⁢ ( Ω , ξ k ) - ( x Ω + ε ⁢ ξ 0 ) ⋅ ξ k : k = { 1 , … , N } ∖ { i 1 , … , i m } } > α 0 2 .

Obviously, for 0 < p < 1 and y 0 , y 0 + △ ⁢ y ∈ ( α 0 2 , + ∞ ) ,

| ( y 0 + △ ⁢ y ) p - y 0 p | < p ⁢ ( α 0 2 ) p - 1 ⁢ | △ ⁢ y | .

From this, the fact that h ⁢ ( Ω , ξ k ) = x Ω ⋅ ξ k , α k > 0 for k ∈ { i 1 , … , i m } , (5.2), (5.3) and (5.4), it follows that

Φ ⁢ ( Ω , x Ω + ε ⁢ ξ 0 ) - Φ ⁢ ( Ω , x Ω ) = ∑ k = 1 N c k ⁢ [ ( h ⁢ ( Ω , ξ k ) - ( x Ω + ε ⁢ ξ 0 ) ⋅ ξ k ) p - ( h ⁢ ( Ω , ξ k ) - x Ω ⋅ ξ k ) p ]

≥ ∑ k ∈ { i 1 , … , i m } c k ( α k ε ) p - ∑ k ∈ { 1 , … , N } ∖ { i 1 , … , i m } c k | ( h ( Ω , ξ k ) - x Ω ⋅ ξ k + α k ε ) p

- ( h ( Ω , ξ k ) - x Ω ⋅ ξ k ) p |

≥ ( ∑ k ∈ { i 1 , … , i m } c k ⁢ α k p ) ⁢ ε p - ∑ k ∈ { 1 , … , N } ∖ { i 1 , … , i m } c k ⁢ p ⁢ ( α 0 2 ) p - 1 ⁢ | α k ⁢ ε |

= ( ∑ k ∈ { i 1 , … , i m } c k ⁢ α k p - ∑ k ∈ { 1 , … , N } ∖ { i 1 , … , i m } c k ⁢ p ⁢ ( α 0 2 ) p - 1 ⁢ | α k | ⁢ ε 1 - p ) ⁢ ε p .

Thus, there exists a small enough ε 0 > 0 such that x Ω + ε 0 ⁢ ξ 0 ∈ Int ⁡ ( Ω ) and

Φ ⁢ ( Ω , x Ω + ε 0 ⁢ ξ 0 ) - Φ ⁢ ( Ω , x Ω ) > 0 .

This contradicts the definition of x Ω . Therefore x Ω ∈ Int ⁡ ( Ω ) . ∎

Lemma 5.2.

Let x Ω , x Ω i be the maximal point of the functional Φ on Ω , Ω i ∈ P ⁢ ( ξ 1 , … , ξ N ) . Suppose Ω i → Ω as i → ∞ . Then x Ω i → x Ω and Φ ⁢ ( Ω i , x Ω i ) → Φ ⁢ ( Ω , x Ω ) as i → ∞ .

Proof.

Since Ω i → Ω as i → ∞ , we have

x Ω i ∈ Ω i ⊆ Ω + B .

This implies { x Ω i } i is a bounded sequence. Let { x Ω i j } j be a convergent subsequence of { x Ω i } i .

Assume { x Ω i j } j → x ′ and x ′ ≠ x Ω . By [34, Theorem 1.8.8], it follows that x ′ ∈ Ω . Hence

Φ ⁢ ( Ω , x ′ ) < Φ ⁢ ( Ω , x Ω ) .

From the continuity of Φ ⁢ ( Ω , x ) in Ω and x, we have

lim j → ∞ ⁡ Φ ⁢ ( Ω i j , x Ω i j ) = Φ ⁢ ( Ω , x ′ ) .

Meanwhile, by [34, Theorem 1.8.8], for x Ω ∈ Ω , there exists a y i j ∈ Ω i j such that y i j → x Ω . Then we have

lim j → ∞ ⁡ Φ ⁢ ( Ω i j , y i j ) = Φ ⁢ ( Ω , x Ω ) .

Hence

(5.5) lim j → ∞ ⁡ Φ ⁢ ( Ω i j , x Ω i j ) < lim j → ∞ ⁡ Φ ⁢ ( Ω i j , y i j ) .

However, for any Ω i j ,

Φ ⁢ ( Ω i j , x Ω i j ) ≥ Φ ⁢ ( Ω i j , y i j ) ,

then we have

lim j → ∞ ⁡ Φ ⁢ ( Ω i j , x Ω i j ) ≥ lim j → ∞ ⁡ Φ ⁢ ( Ω i j , y i j ) ,

which contradicts (5.5). Thus, x Ω i j → x Ω , and x Ω i → x Ω . From the continuity of Φ, then

Φ ⁢ ( Ω i , x Ω i ) → Φ ⁢ ( Ω , x Ω ) .

This completes the proof. ∎

Lemma 5.3.

Suppose Ω ∈ P ⁢ ( ξ 1 , … , ξ N ) . Then:

Φ ⁢ ( Ω + y , x Ω + y ) = Φ ⁢ ( Ω , x Ω ) , for y ∈ ℝ n .
Φ ⁢ ( λ ⁢ Ω , x λ ⁢ Ω ) = λ p ⁢ Φ ⁢ ( Ω , x Ω ) , for λ > 0 .

Proof.

From (5.1), we have

Φ ⁢ ( Ω + y , x Ω + y ) = max z ∈ Ω + y ⁡ Φ ⁢ ( Ω + y , z )

= max z - y ∈ Ω ⁢ ∑ k = 1 N c k ⁢ ( h ⁢ ( Ω + y , ξ k ) - z ⋅ ξ k ) p

= max z - y ∈ Ω ⁢ ∑ k = 1 N c k ⁢ ( h ⁢ ( Ω , ξ k ) - ( z - y ) ⋅ ξ k ) p

= max x ∈ Ω ⁢ ∑ k = 1 N c k ⁢ ( h ⁢ ( Ω , ξ k ) - x ⋅ ξ k ) p

= Φ ⁢ ( Ω , x Ω ) .

In the same way, we can get a proof of (2). ∎

5.2 An extremal problem and the L p Minkowski problem

Suppose 0 1 . In the following, we study the extremal problem

(5.6) inf ⁡ { max x ∈ Ω ⁡ Φ ⁢ ( Ω , x ) : Ω ∈ 𝒫 ⁢ ( ξ 1 , … , ξ N ) , T q ⁢ ( Ω ) = 1 } ,

and show that its solution is exactly the solution of the L p Minkowski problem for the q-torsional rigidity we are concerned with.

Lemma 5.4.

Suppose P ∈ P ⁢ ( ξ 1 , … , ξ N ) with normal vectors ξ 1 , … , ξ N . If P is the solution to problem (5.6), and x P = o , then

λ ⁢ h ⁢ ( P , ⋅ ) 1 - p ⁢ d ⁢ μ q tor ⁢ ( P , ⋅ ) = d ⁢ μ ,

where λ = q - 1 q + n ⁢ ( q - 1 ) ⁢ ∑ k = 1 N c k ⁢ h ⁢ ( P , ξ k ) p .

Proof.

For δ 1 , … , δ N > 0 and sufficiently small | t | > 0 , Let

P t = { x : x ⋅ ξ k ≤ h ⁢ ( P , ξ k ) + t ⁢ δ k , k = 1 , … , N }

and

γ ⁢ ( t ) ⁢ P t = T q ⁢ ( P t ) - 1 q + n ⁢ ( q - 1 ) ⁢ P t .

Then T q ⁢ ( γ ⁢ ( t ) ⁢ P t ) = 1 , γ ⁢ ( t ) ⁢ P t ∈ 𝒫 N ⁢ ( ξ 1 , … , ξ N ) and γ ⁢ ( t ) ⁢ P t → P as t → 0 .

We denote x ⁢ ( t ) = x γ ⁢ ( t ) ⁢ P t . Let

(5.7) Φ ⁢ ( γ ⁢ ( t ) ⁢ P t , x ⁢ ( t ) ) = max x ∈ γ ⁢ ( t ) ⁢ P t ⁢ ∑ k = 1 N c k ⁢ ( γ ⁢ ( t ) ⁢ h ⁢ ( P t , ξ k ) - x ⋅ ξ k ) p = ∑ k = 1 N c k ⁢ ( γ ⁢ ( t ) ⁢ h ⁢ ( P t , ξ k ) - x ⁢ ( t ) ⋅ ξ k ) p .

Since x ⁢ ( t ) is an interior point of γ ⁢ ( t ) ⁢ P t , by (5.7), we have

∑ k = 1 N c k ⁢ ξ k , i [ γ ⁢ ( t ) ⁢ h ⁢ ( P t , ξ k ) - x ⁢ ( t ) ⋅ ξ k ] 1 - p = 0 , i = 1 , … , n ,

where ξ k = ( ξ k , 1 , … , ξ k , n ) ⊤ . Let t = 0 . Then P 0 = P , γ ⁢ ( 0 ) = 1 , x ⁢ ( 0 ) = o and

(5.8) ∑ k = 1 N c k ⁢ ξ k , i h ⁢ ( P , ξ k ) 1 - p = 0 , i = 1 , … , n .

Therefore

(5.9) ∑ k = 1 N c k ⁢ ξ k h ⁢ ( P , ξ k ) 1 - p = 0 .

Now we need to show x ′ ⁢ ( t ) | t = 0 exists. Let

y i ⁢ ( t , x 1 , … , x n ) = ∑ k = 1 N c k ⁢ ξ k , i [ γ ⁢ ( t ) ⁢ h ⁢ ( P t , ξ k ) - ( x 1 ⁢ ξ k , 1 + ⋯ + x n ⁢ ξ k , n ) ] 1 - p ,

for i = 1 , … , n . Then

∂ ⁡ y i ∂ ⁡ x j | 0 , … , 0 = ∑ k = 1 N ( 1 - p ) ⁢ c k h ⁢ ( P , ξ k ) 2 - p ⁢ ξ k , i ⁢ ξ k , j .

Thus

( ∂ ⁡ y ∂ ⁡ x | 0 , … , 0 ) n × n = ∑ k = 1 N ( 1 - p ) ⁢ c k h ⁢ ( P , ξ k ) 2 - p ⁢ ξ k ⁢ ξ k ⊤ .

For x ∈ ℝ n with x ≠ 0 . Since ξ 1 , … , ξ N are not concentrated on any closed hemisphere, there exists an element ξ i 0 ∈ { ξ 1 , … , ξ N } such that ξ i 0 ⋅ x ≠ 0 . Thus

x ⊤ ⁢ ( ∑ k = 1 N ( 1 - p ) ⁢ c k h ⁢ ( P , ξ k ) 2 - p ⁢ ξ k ⁢ ξ k ⊤ ) ⁢ x = ∑ k = 1 N ( 1 - p ) ⁢ c k h ⁢ ( P , ξ k ) 2 - p ⁢ ( x ⋅ ξ k ) 2 ≥ ( 1 - p ) ⁢ c i 0 h ⁢ ( P , ξ i 0 ) 2 - p ⁢ ( x ⋅ ξ i 0 ) 2 > 0 ,

which implies that ( ∂ ⁡ y ∂ ⁡ x | 0 , … , 0 ) n × n is positive definite. This, combining (5.8) and the inverse function theorem, it follows that x ′ ⁢ ( 0 ) = ( x 1 ′ ⁢ ( 0 ) , … , x n ′ ⁢ ( 0 ) ) exists.

Next, we can finish the proof. Since the functional Φ attains its minimum at the polytope P, From (3.3) and (5.9), we have

0 = 1 p ⁢ d ⁢ Φ ⁢ ( γ ⁢ ( t ) ⁢ P t , x ⁢ ( t ) ) d ⁢ t | t = 0

= ∑ j = 1 N c j ⁢ h ⁢ ( P , ξ j ) p - 1 ⁢ [ h ⁢ ( P , ξ j ) ⁢ ( - 1 q + n ⁢ ( q - 1 ) ) ⁢ d ⁢ T q ⁢ ( P t ) d ⁢ t | t = 0 + δ j - x ′ ⁢ ( 0 ) ⋅ ξ j ]

= ∑ j = 1 N c j ⁢ h ⁢ ( P , ξ j ) p - 1 ⁢ [ h ⁢ ( P , ξ j ) ⁢ ( - q - 1 q + n ⁢ ( q - 1 ) ) ⁢ ( ∑ k = 1 N δ k ⁢ μ q tor ⁢ ( P , { ξ k } ) ) + δ j ] - x ′ ⁢ ( 0 ) ⁢ ( ∑ j = 1 N c j ⁢ h ⁢ ( P , ξ j ) p - 1 ⁢ ξ j )

= ∑ j = 1 N c j ⁢ h ⁢ ( P , ξ j ) p - 1 ⁢ [ h ⁢ ( P , ξ j ) ⁢ ( - q - 1 q + n ⁢ ( q - 1 ) ) ⁢ ( ∑ k = 1 N δ k ⁢ μ q tor ⁢ ( P , { ξ k } ) ) + δ j ]

= ∑ j = 1 N δ j ⁢ [ c j ⁢ h ⁢ ( P , ξ j ) p - 1 - ( q - 1 q + n ⁢ ( q - 1 ) ) ⁢ ( ∑ k = 1 N c k ⁢ h ⁢ ( P , ξ k ) p ) ⁢ μ q tor ⁢ ( P , { ξ k } ) ] .

For arbitrary positive real numbers δ 1 , … , δ N , we have

q - 1 q + n ⁢ ( q - 1 ) ⁢ ( ∑ k = 1 N c k ⁢ h ⁢ ( P , ξ k ) p ) ⁢ μ q tor ⁢ ( P , { ξ k } ) = c j ⁢ h ⁢ ( P , ξ j ) p - 1 ,

for j = 1 , … , N . Since P is containing the origin o in its interior, we have h ⁢ ( P , ξ j ) > 0 , and thus we know

q - 1 q + n ⁢ ( q - 1 ) ⁢ ( ∑ k = 1 N c k ⁢ h ⁢ ( P , ξ k ) p ) ⁢ h ⁢ ( P , ξ j ) 1 - p ⁢ μ q tor ⁢ ( P , { ξ k } ) = c j ,

for j = 1 , … , N . Therefore,

λ ⁢ h ⁢ ( P , ⋅ ) 1 - p ⁢ d ⁢ μ q tor ⁢ ( P , ⋅ ) = d ⁢ μ ,

where λ = q - 1 q + n ⁢ ( q - 1 ) ⁢ ∑ k = 1 N c k ⁢ h ⁢ ( P , ξ k ) p . ∎

The following lemma shows that the solution to (5.6) is just a scaling of the solution to the L p Minkowski problem for the q-torsional rigidity.

Lemma 5.5.

Suppose P ∈ P ⁢ ( ξ 1 , … , ξ N ) with normal vector ξ 1 , … , ξ N . If P is the solution to problem (5.6), and x P = o , then

d ⁢ μ p , q tor ⁢ ( λ 0 ⁢ P , ⋅ ) = d ⁢ μ for ⁢ λ 0 = ( q - 1 q + n ⁢ ( q - 1 ) ⁢ ∑ k = 1 N c k ⁢ h ⁢ ( P , ξ k ) p ) 1 q + n ⁢ ( q - 1 ) - p .

Proof.

Let s > 0 and P ∈ 𝒫 ⁢ ( ξ 1 , … , ξ N ) . Then

(5.10) d ⁢ μ p , q tor ⁢ ( s ⁢ P , ⋅ ) = s q + n ⁢ ( q - 1 ) - p ⁢ h ⁢ ( P , ⋅ ) 1 - p ⁢ d ⁢ μ q ⁢ ( P , ⋅ ) = s q + n ⁢ ( q - 1 ) - p ⁢ d ⁢ μ p , q tor ⁢ ( P , ⋅ ) .

Since 0 1 , we have n ⁢ ( q - 1 ) ≠ p - q . If P is the solution to (5.6), then, by Lemma 5.4, we have

λ ⁢ d ⁢ μ p , q tor ⁢ ( P , ⋅ ) = λ ⁢ h ⁢ ( P , ⋅ ) 1 - p ⁢ d ⁢ μ q ⁢ ( P , ⋅ ) = d ⁢ μ ,

where λ = q - 1 q + n ⁢ ( q - 1 ) ⁢ ∑ k = 1 N c k ⁢ h ⁢ ( P , ξ k ) p . This together with (5.10) implies

d ⁢ μ p , q tor ⁢ ( λ 0 ⁢ P , ⋅ ) = d ⁢ μ ,

where λ 0 = λ 1 q + n ⁢ ( q - 1 ) - p . This completes the proof. ∎

5.3 Existence of solutions to the L p Minkowski problem for q-torsional rigidity

We also need the following two lemmas to complete the existence of solutions to the L p Minkowski problem of the q-torsional rigidity for discrete measures when 0 1 .

Lemma 5.6.

Suppose P ∈ P ⁢ ( ξ 1 , … , ξ N ) with normal vectors ξ 1 , … , ξ N , and 0 < p < 1 . If P is the solution to Problem (5.6), and x P = o . Then P has exactly N facets whose normal vectors are ξ 1 , … , ξ N .

Proof.

We now argue Lemma 5.6 by reductio. To this end, assume that ξ i 0 ∈ { ξ 1 , … , ξ N } , but the support set ℱ ⁢ ( P , ξ i 0 ) = P ∩ H ⁢ ( P , ξ i 0 ) is not a facet of P.

Fix δ > 0 , let

P δ = P ∩ { x : x ⋅ ξ i 0 ≤ h ⁢ ( P , ξ i 0 ) - δ } ∈ 𝒫 ⁢ ( ξ 1 , … , ξ N ) ,

and

τ ⁢ P δ = τ ⁢ ( δ ) ⁢ P δ = T q ⁢ ( P δ ) - 1 q + n ⁢ ( q - 1 ) ⁢ P δ .

Then T q ⁢ ( τ ⁢ P δ ) = 1 and τ ⁢ P δ → P as δ → 0 + . By Lemma 5.2, we see that x P δ → x P = o ∈ int ( P ) as δ → 0 + . Thus, for sufficiently small δ > 0 , we can assume that x P δ ∈ Int ⁡ ( P ) and

h ⁢ ( P , ξ k ) - x P δ ⋅ ξ k > δ > 0 , k = 1 , … , N .

In the following, we show Φ ⁢ ( τ ⁢ P δ , x τ ⁢ P δ ) < Φ ⁢ ( P , o ) , which contradicts the fact that Φ ⁢ ( P , o ) is the minimum. We have

Φ ⁢ ( τ ⁢ P δ , x τ ⁢ P δ ) = τ p ⁢ ∑ k = 1 N c k ⁢ ( h ⁢ ( P δ , ξ k ) - x P δ ⋅ ξ k ) p

= τ p ⁢ ( ∑ k = 1 N c k ⁢ ( h ⁢ ( P , ξ k ) - x P δ ⋅ ξ k ) p ) + τ p ⁢ c i 0 ⁢ ( h ⁢ ( P , ξ i 0 ) - x P δ ⋅ ξ i 0 - δ ) p - τ p ⁢ c i 0 ⁢ ( h ⁢ ( P , ξ i 0 ) - x P δ ⋅ ξ i 0 ) p

= Φ ⁢ ( P , x P δ ) + G ⁢ ( δ ) ,

where

G ⁢ ( δ ) = ( τ p - 1 ) ⁢ ( ∑ k = 1 N c k ⁢ ( h ⁢ ( P , ξ k ) - x P δ ⋅ ξ k ) p ) + c i 0 ⁢ τ p ⁢ [ ( h ⁢ ( P , ξ i 0 ) - x P δ ⋅ ξ i 0 - δ ) p - ( h ⁢ ( P , ξ i 0 ) - x P δ ⋅ ξ i 0 ) p ] .

If we can prove G ⁢ ( δ ) < 0 , then Φ ⁢ ( τ ⁢ P δ , x τ ⁢ P δ ) < Φ ⁢ ( P , x P δ ) ≤ Φ ⁢ ( P , o ) , as desired.

Since 0 < h ⁢ ( P , ξ i 0 ) - x P δ ⋅ ξ i 0 - δ < h ⁢ ( P , ξ i 0 ) - x P δ ⋅ ξ i 0 < d 0 , where d 0 is the diameter of P, by the concavity of t p on [ 0 , ∞ ) for 0 < p < 1 , it follows that

( h ⁢ ( P , ξ i 0 ) - x P δ ⋅ ξ i 0 - δ ) p - ( h ⁢ ( P , ξ i 0 ) - x P δ ⋅ ξ i 0 ) p < ( d 0 - δ ) p - d 0 p .

Hence

G ⁢ ( δ ) < ( τ p - 1 ) ⁢ ( ∑ k = 1 N c k ⁢ ( h ⁢ ( P , ξ k ) - x P δ ⋅ ξ k ) p ) + c i 0 ⁢ τ p ⁢ ( ( d 0 - δ ) p - d 0 p )

= τ p ⁢ ( ( d 0 - δ ) p - d 0 p ) ⁢ ( c i 0 + 1 τ p ⁢ τ p - 1 ( d 0 - δ ) p - d 0 p ⁢ ∑ k = 1 N c k ⁢ ( h ⁢ ( P , ξ k ) - x P δ ⋅ ξ k ) p ) .

From formula (3.3) and T q ⁢ ( P ) = 1 , we have

lim δ → 0 + ⁡ τ p - 1 ( d 0 - δ ) p - d 0 p = lim δ → 0 + ⁡ T q ⁢ ( P δ ) - p q + n ⁢ ( q - 1 ) - 1 ( d 0 - δ ) p - d 0 p

= - p ⁢ ( q - 1 ) q + n ⁢ ( q - 1 ) ⁢ ∑ k = 1 N μ q tor ⁢ ( P , { ξ k } ) ⁢ h ′ ⁢ ( ξ k , 0 ) - p ⁢ d 0 p - 1

= ( q - 1 ) q + n ⁢ ( q - 1 ) ⁢ ∑ k = 1 N μ q tor ⁢ ( P , { ξ k } ) ⁢ h ′ ⁢ ( ξ k , 0 ) d 0 p - 1 ,

where h ′ ⁢ ( ξ k , 0 ) = lim δ → 0 + ⁡ h ⁢ ( P δ , ξ k ) - h ⁢ ( P , ξ k ) δ .

Assume μ q tor ⁢ ( P , { ξ k } ) ≠ 0 for some k. Since μ q tor ⁢ ( P , ⋅ ) is absolutely continuous with respect to surface area measure S ⁢ ( P , ⋅ ) , it follows that P has a facet with normal vector ξ k . By the definition of P δ , we have h ⁢ ( P δ , ξ k ) = h ⁢ ( P , ξ k ) for sufficiently small δ > 0 . Thus h ′ ⁢ ( ξ k , 0 ) = 0 and

∑ k = 1 N μ q tor ⁢ ( P , { ξ k } ) ⁢ h ′ ⁢ ( ξ k , 0 ) = 0 .

Therefore,

lim δ → 0 + ⁡ τ p - 1 ( d 0 - δ ) p - d 0 p = 0 .

This together with ( d 0 - δ ) p - d 0 p < 0 , c i 0 > 0 and

1 τ p ⁢ ∑ k = 1 N c k ⁢ ( h ⁢ ( P , ξ k ) - x P δ ⋅ ξ k ) → ∑ k = 1 N c k ⁢ h ⁢ ( P , ξ k ) p > 0 ,

as δ → 0 + , implies that, for sufficiently small δ > 0 , G ⁢ ( δ ) < 0 .

Consequently, P has exactly N facets. This completes the proof. ∎

Lemma 5.7.

Suppose μ is a finite positive Borel measure on S n - 1 which is not concentrated on any closed hemisphere. Then, for μ = ∑ k = 1 N c k ⁢ δ ξ k , there exists a polytope P solving problem (5.6).

Proof.

Let

β = inf ⁡ { max x ∈ Ω ⁡ Φ ⁢ ( Ω , x ) : Ω ∈ 𝒫 ⁢ ( ξ 1 , … , ξ N ) , T q ⁢ ( Ω ) = 1 } .

Take a minimizing sequence { P i } i such that P i ∈ 𝒫 ⁢ ( ξ 1 , … , ξ N ) , x P i = o , T q ⁢ ( P i ) = 1 and lim i → ∞ ⁡ Φ ⁢ ( P i , o ) = β .

Next, we prove that { P i } i is bounded. Since x P = o , by the definition of Φ, it follows that

∑ k = 1 N c k ⁢ h ⁢ ( P i , ξ k ) p = max x ∈ P i ⁢ ∑ k = 1 N c k ⁢ ( h ⁢ ( P i , ξ k ) - x ⋅ ξ k ) p

≤ max x ∈ τ ⁢ Ω ⁢ ∑ k = 1 N c k ⁢ ( h ⁢ ( τ ⁢ Ω , ξ k ) - x ⋅ ξ k ) p + 1 ,

where Ω = { x : x ⋅ ξ k ≤ 1 , k = 1 , … , N } and τ satisfies T q ⁢ ( τ ⁢ Ω ) = 1 . Let

ℳ = max x ∈ τ ⁢ Ω ⁢ ∑ k = 1 N c k ⁢ ( h ⁢ ( τ ⁢ Ω , ξ k ) - x ⋅ ξ k ) p + 1 .

Then ℳ > 0 is independent of i. Hence, for any i,

h ⁢ ( P i , ξ k ) ≤ ( ℳ min 1 ≤ k ≤ N ⁡ c k ) 1 p < ∞ , k = 1 , … , N ,

which implies that { P i } i is bounded.

By Lemma 5.2 and the Blaschke selection theorem, there exists a convergent subsequence { P i j } j of { P i } i such that P i j → P . ∎

Finally, we give the existence of solution to the L p Minkowski problem of the q-torsional rigidity for discrete measure when 0 1 .

Theorem 5.8.

Let μ be a finite positive Borel measure on S n - 1 which is not concentrated on any closed hemisphere. If 0 1 , then, for μ = ∑ k = 1 N c k ⁢ δ ξ k , there exists a polytope P containing the origin in its interior such that μ p , q tor ⁢ ( P , ⋅ ) = μ .

Proof.

For the discrete measure μ, by Lemma 5.7, there exists a polytope Ω 0 which solves problem (5.6), that is, T q ⁢ ( Ω 0 ) = 1 and

Φ ⁢ ( Ω 0 , x Ω 0 ) = inf ⁡ { max x ∈ Ω ⁡ Φ ⁢ ( Ω , x ) : Ω ∈ 𝒫 ⁢ ( ξ 1 , … , ξ N ) , T q ⁢ ( Ω ) = 1 } .

By Lemma 5.3, then P 0 = Ω 0 - x Ω 0 is still the solution to problem (5.6) and x P 0 = o . This together with Lemma 5.6, Lemma 5.4 and Lemma 5.5, we have

μ p , q ⁢ ( λ 0 ⁢ P 0 , ⋅ ) = μ ,

where λ 0 = ( q - 1 q + n ⁢ ( q - 1 ) ⁢ ∑ k = 1 N c k ⁢ h ⁢ ( P , ξ k ) p ) 1 q + n ⁢ ( q - 1 ) - p . Namely P = λ 0 ⁢ P 0 is the desired solution. ∎

6 The L p Minkowski problem of q-torsional rigidity for general measure

Let Ω be an open bounded convex set of ℝ n and let φ be the solution of (1.3) in Ω. Let M Ω = max Ω ¯ ⁡ φ , for every t ∈ [ 0 , M Ω ] , we define

Ω t = { x ∈ Ω : φ ⁢ ( x ) > t } .

By the general concavity theorems in [23, 24], Ω t is convex for every t. Moreover, we have ∇ ⁡ φ = 0 if and only if φ ⁢ ( x ) = M Ω such that

∂ ⁡ Ω t = { x ∈ Ω : φ ⁢ ( x ) = t } , t ∈ ( 0 , M Ω ) .

The following lemma shows an L ∞ estimate for the gradient of φ.

Lemma 6.1.

Let Ω be an open bounded convex subset of R n and let φ be the solution of problem (1.3) in Ω. Then, for every x ∈ Ω ,

| ∇ ⁡ φ ⁢ ( x ) | ≤ diam ⁡ ( Ω ) .

Proof.

Let x ′ ∈ Ω and t = φ ⁢ ( x ′ ) > 0 . If φ ⁢ ( x ′ ) = M Ω , then ∇ ⁡ φ ⁢ ( x ′ ) = 0 and the claim is true.

Assume φ ⁢ ( x ′ ) < M Ω . This implies that x ′ ∈ ∂ ⁡ Ω t . The convex set Ω t admits a support hyperplane H at x ′ . We may choose an orthogonal coordinate system with origin o and coordinates x 1 , … , x n , in ℝ n , such that x ′ = o , H = { x ∈ ℝ n : x n = 0 } and Ω t ⊂ { x ∈ ℝ n : x n ≥ 0 } . By a standard argument based on the implicit function theorem, ∂ ⁡ M t is of class C ∞ such that H is in fact the tangent hyperplane to ∂ ⁡ M t at x ′ . Consequently, we have

| ∇ ⁡ φ ⁢ ( x ′ ) | = ∂ ⁡ φ ∂ ⁡ x n ⁢ ( x ′ ) .

We also have the inclusion Ω t ⊂ { x ∈ ℝ n : x n ≤ d } , where d = diam ⁡ ( Ω ) . Let us introduce the function

ψ ⁢ ( x ) = ψ ⁢ ( x 1 , … , x n ) = t + x n ⁢ ( d - x n ) , x ∈ ℝ n .

Note that △ q ⁢ ψ ⁢ ( x ) = - 1 for every x ∈ ℝ n and ψ ⁢ ( x ) ≥ t for x ∈ { x = ( x 1 , … , x n ) ∈ ℝ n : 0 ≤ x n ≤ d } . In particular, ψ ≥ φ on ∂ ⁡ Ω t , and by the comparison principle,

ψ ⁢ ( x ) ≥ φ ⁢ ( x ) , x ∈ Ω t .

Finally, as φ ⁢ ( x ′ ) = ψ ⁢ ( x ′ ) , we have

∂ ⁡ φ ∂ ⁡ x n ⁢ ( x ′ ) ≤ ∂ ⁡ ψ ∂ ⁡ x n ⁢ ( x ′ ) = d . ∎

Theorem 6.2.

Let μ be a finite positive Borel measure on S n - 1 which is not concentrated on any closed hemisphere. If 0 1 , then there exists a convex body K ∈ K o n such that

μ p , q tor ⁢ ( K , ⋅ ) = μ .

Proof.

Let the given measure μ satisfy the assumptions in Theorem 6.2. Then there exists a sequence of discrete measures μ i defined on 𝕊 n - 1 whose support is not contained in any closed hemisphere so that μ i → μ weakly as i → ∞ (see [34, proof of Theorem 7.1.2]).

From Theorem 5.8, for each μ i , there are polytopes P i containing the origin in their interior such that, for i ≥ 1 ,

(6.1) μ i = h ⁢ ( P i , ⋅ ) 1 - p ⁢ μ q tor ⁢ ( P i , ⋅ ) = μ p , q tor ⁢ ( P i , ⋅ ) .

Now we show that the sequence of { P i } is bounded. Let R i := max ⁡ { h ⁢ ( P i , v ) : v ∈ 𝕊 n - 1 } and choose v 0 ∈ 𝕊 n - 1 such that R i = h ⁢ ( P i , v 0 ) . Then [ 0 , R i ⁢ v i ] ⊂ P i , and thus R i ⁢ 〈 u , v 0 〉 + ≤ h ⁢ ( P i , u ) for u ∈ 𝕊 n - 1 . Hence, by the argument of [21, Lemma 2.3], we have

R i ≤ c 0 ,

for some constant c 0 > 0 . This shows that the sequence of { P i } is bounded. By the Blaschke selection theorem, there exists a subsequence of { P i } which converges to a compact convex set K. Obviously, o ∈ K since o ∈ P i . From Lemma 6.1 and the definition of T q ⁢ ( K ) , we have

V ⁢ ( P i ) q - 1 ≥ 1 ( diam ⁡ ( P i ) ) q ⁢ ( q - 1 ) ⁢ T q ⁢ ( P i ) ≥ 1 d q ⁢ ( q - 1 ) ⁢ T q ⁢ ( B 2 n ) > 0 ,

for some d > 0 . This implies that V ⁢ ( K ) > 0 due to lim i → ∞ ⁡ P i = K . Thus K is a convex body containing the origin in its interior.

Next, we show that the K is the required solution. For any continuous function f ∈ C ⁢ ( 𝕊 n - 1 ) , by (6.1), one has

∫ 𝕊 n - 1 f ⁢ ( u ) h ⁢ ( P i , u ) 1 - p ⁢ 𝑑 μ i = ∫ 𝕊 n - 1 f ⁢ ( u ) ⁢ 𝑑 μ q tor ⁢ ( P i , u ) .

Since h ⁢ ( P i , ⋅ ) → h ⁢ ( K , ⋅ ) uniformly on 𝒮 n - 1 , μ i → μ weakly as i → ∞ , and μ q tor ⁢ ( P i , ⋅ ) → μ q tor ⁢ ( K , ⋅ ) weakly as i → ∞ , it follows that there is

∫ 𝕊 n - 1 f ⁢ ( u ) h ⁢ ( K , u ) 1 - p ⁢ 𝑑 μ = ∫ 𝕊 n - 1 f ⁢ ( u ) ⁢ 𝑑 μ q tor ⁢ ( K , u )

Since f ∈ C ⁢ ( 𝕊 n - 1 ) is arbitrary, we have

μ = μ p , q tor ⁢ ( K , ⋅ ) .

This ends the proof of Theorem 6.2. ∎

Communicated by Guofang Wang

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12271254

Award Identifier / Grant number: 12141104

Funding statement: Peibiao Zhao was supported by the Natural Science Foundation of China (No. 12271254, No. 12141104).

Acknowledgements

The authors would like to express their heartfelt thanks to Professor J. Zhou and Professor H. Li for helpful comments and suggestions.

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Received: 2022-05-30

Accepted: 2022-10-18

Published Online: 2022-11-24

Published in Print: 2024-07-01

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

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https://doi.org/10.1515/acv-2022-0041

Keywords for this article

Minkowski problem; Minkowski inequality

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The Lp Minkowski problem for q-torsional rigidity

Article

Abstract

1 Introduction

Remark 1.1.

Theorem 1.2.

Definition 1.3.

Problem 1.4 ( L p Minkowski problem of q-torsional rigidity).

Theorem 1.5.

Theorem 1.6.

2 Preliminaries

2.1 Basic facts on convex body

2.2 Aleksandrov body

Aleksandrov’s Convergence Lemma.

3 Variational formula for the q-torsional rigidity of general convex bodies

Lemma 3.1 ([22]).

Lemma 3.2.

Theorem 3.3.

Proof.

Theorem 3.4.

4 The L p q-torsional measure

Theorem 4.1.

Proof.

Definition 4.2.

Definition 4.3.

Lemma 4.4.

Theorem 4.5.

Theorem 4.6.

5 The L p Minkowski problem of the q-torsional rigidity for discrete measures

5.1 An extremal problem

Lemma 5.1.

Proof.

Lemma 5.2.

Proof.

Lemma 5.3.

Proof.

5.2 An extremal problem and the L p Minkowski problem

Lemma 5.4.

Proof.

Lemma 5.5.

Proof.

5.3 Existence of solutions to the L p Minkowski problem for q-torsional rigidity

Lemma 5.6.

Proof.

Lemma 5.7.

Proof.

Theorem 5.8.

Proof.

6 The L p Minkowski problem of q-torsional rigidity for general measure

Lemma 6.1.

Proof.

Theorem 6.2.

Proof.

Acknowledgements

References

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The L_p Minkowski problem for q-torsional rigidity