The Lp chord Minkowski problem

Dongmeng Xi; Deane Yang; Gaoyong Zhang; Yiming Zhao

doi:10.1515/ans-2022-0041

Article Open Access

The L_p chord Minkowski problem

Dongmeng Xi , Deane Yang , Gaoyong Zhang and Yiming Zhao

Published/Copyright: January 13, 2023

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From the journal Advanced Nonlinear Studies Volume 23 Issue 1

Abstract

Chord measures are newly discovered translation-invariant geometric measures of convex bodies in R n , in addition to Aleksandrov-Fenchel-Jessen’s area measures. They are constructed from chord integrals of convex bodies and random lines. Prescribing the L p chord measures is called the L p chord Minkowski problem in the L p Brunn-Minkowski theory, which includes the L p Minkowski problem as a special case. This article solves the L p chord Minkowski problem when p > 1 and the symmetric case of 0 < p < 1 .

Keywords: chord integral; chord measure; L p surface area measure; L p chord measure; L p Minkowski problem; L p chord Minkowski problem

MSC 2010: 52A38

1 Introduction

The classical Minkowski problem asks for the existence, uniqueness, and regularity of a convex body whose surface area measure is equal to a pre-given spherical Borel measure. When the given measure has a positive continuous density, the Minkowski problem is also known as the problem of prescribing the Gauss curvature in differential geometry. For other important geometric measures of convex bodies, similar Minkowski problems have been studied in convex geometry, differential geometry, and partial differential equations. The study of Minkowski problems has motivated the study of fully nonlinear partial differential equations, geometric curvature flows, and geometric inequalities.

The surface area measure of a convex body in Euclidean space is a Borel measure on the unit sphere, which was introduced by Aleksandrov-Fenchel-Jessen in the 1930s. It is the differential of the volume functional over convex bodies. Similar concepts for surface area and other quermassintegrals, called area measures, were also introduced by them. Area measures are translation invariant. Another family of geometric measures associated with quermassintegrals is the curvature measures of Federer. Area measures and curvature measures are fundamental concepts in the classical Brunn-Minkowski theory.

In the 1970s, Lutwak introduced the dual Brunn-Minkowski theory [34]. The duality between projections and intersections of convex bodies and their connections with harmonic analysis were the focus in the 1990s. Significant breakthroughs were made, see, for example, [7,10,15,29,35,54], and the books of Gardner [11] and Koldobsky [30]. However, what acts as the dual counterpart of the geometric measures in the Brunn-Minkowski theory was not clear until the work of Huang et al. [23] in 2016. Their discovery of dual curvature measures gives dual concepts to Federer’s curvature measures. The dual Minkowski problem posed by them has led to a number of articles in a short period of time, see, for example, [6,13,14,25,26,31,45,57,58]. Dual curvature measures are differentials of the dual quermassintegrals. Dual quermassintegrals are origin dependent and thus not translation invariant. Their translation invariant analogues are chord integrals, which are basic geometric invariants in integral geometry [46,47,53].

Very recently, Lutwak et al. [38] constructed the geometric measures, which are the differentials of chord integrals and then called chord measures. Chord measures are translation invariant-like area measures. Minkowski problems associated with chord measures were posed in [38]. These geometric problems give new Monge-Ampère-type partial differential equations. The chord Minkowski problem includes the classical Minkowski problem as a special case and an unsolved Christoffel-Minkowski problem as a critical case, while the chord log-Minkowski problem includes the logarithmic Minkowski problem as an important case. Thus, the new chord Minkowski problems greatly enrich the area of Minkowski problems. Xi et al. [38] solved the chord Minkowski problem, except for the critical case of the Christoffel-Minkowski problem and partially solved the symmetric case of the chord log-Minkowski problem. They also posed the more general L p chord Minkowski problem, which includes the L p Minkowski problem as a special case. This article solves the L p chord Minkowski problem when p > 1 and the symmetric case of 0 < p < 1 . These results generalize known results for the corresponding cases of the L p Minkowski problem.

Let K n be the collection of convex bodies (compact convex sets with nonempty interior) in R n . For K ∈ K n , the chord integral I q ( K ) of K is defined as follows:

I q ( K ) = ∫ L n ∣ K ∩ ℓ ∣ q d ℓ , q ≥ 0 ,

where ∣ K ∩ ℓ ∣ denotes the length of the chord K ∩ ℓ , and the integration is with respect to the (appropriately normalized) Haar measure on the affine Grassmannian L n of lines in R n . Chord integrals contain volume V ( K ) and surface area S ( K ) as two important special cases:

I 1 ( K ) = V ( K ) , I 0 ( K ) = ω n − 1 n ω n S ( K ) , I n + 1 ( K ) = n + 1 ω n V ( K ) 2 ,

where ω n is the volume enclosed by the unit sphere S n − 1 .

It was shown in [38] that the differential of I q ( K ) defines a finite Borel measure F q ( K , ⋅ ) on S n − 1 . Precisely, for convex bodies K and L in R n , we have

(1.1) d d t t = 0 + I q ( K + t L ) = ∫ S n − 1 h L ( v ) d F q ( K , v ) , q ≥ 0 ,

where F q ( K , ⋅ ) is called the qth chord measure of K and h L is the support function of L . The cases of q = 0 , 1 of this formula are classical, which are the variational formulas of surface area and volume. There are

F 0 ( K , ⋅ ) = ( n − 1 ) ω n − 1 n ω n S n − 2 ( K , ⋅ ) , F 1 ( K , ⋅ ) = S n − 1 ( K , ⋅ ) ,

where S n − 2 ( K , ⋅ ) is the ( n − 2 ) th order area measure of K , and S n − 1 ( K , ⋅ ) is the ( n − 1 ) th order area measure of K (i.e., the classical surface area measure of K ).

The chord Minkowski problem states:

If μ is a finite Borel measure on S n − 1 , what are the necessary and sufficient conditions for the existence of a convex body K that solves the equation,

F q ( K , ⋅ ) = μ ?

This is a new Minkowski problem except q = 0 , 1 . The case of q = 1 is the classical Minkowski problem for surface area measure, and the case of q = 0 is the unsolved Christoffel-Minkowski problem for the ( n − 2 ) th area measure. When q > 0 , the solution to the chord Minkowski problem was given in [38].

The L p Minkowski problem was posed by Lutwak [36] in the early 1990s. He defined the fundamental concept of L p surface area measure in the L p Brunn-Minkowski theory, which has led to fruitful studies in several areas, including affine isoperimetric and Sobolev inequalities [8,19,20, 21,22,39,41,55], affine surface areas and valuations [27,32,33,44,50,51,56], and Minkowski problems [9,17,24,26,28,42]. The Minkowski problem of prescribing L p surface area measures is the L p Minkowski problem. Many cases of the L p Minkowski problem have been solved. However, critical cases of the problem and uniqueness for p < 1 remain open. In particular, the centro-affine Minkowski problem [9] and the logarithmic Brunn-Minkowski conjecture [2] are highly interesting.

Denote by K o n the sub-collection of K n of convex bodies that contain the origin in their interiors, and by K e n the sub-collection of K n of convex bodies that are symmetric about the origin. The L p surface area measure can be extended to a two-parameter family of geometric measures, called L p chord measures. The (p,q)th chord measure, F p , q ( K , ⋅ ) of K ∈ K o n is defined as follows:

d F p , q ( K , ⋅ ) = h K 1 − p d F q ( K , ⋅ ) , p ∈ R , q ≥ 0 ,

where h K is the support function of K and F q ( K , ⋅ ) is the q th chord measure of K . When q = 1 , F p , 1 ( K , ⋅ ) is the L p surface area measure. When q = 0 , F p , 0 ( K , ⋅ ) is the L p ( n − 2 ) th area measure. When p = 1 , F 1 , q ( K , ⋅ ) is just the q th chord measure F q ( K , ⋅ ) .

The L p chord Minkowski problem asks:

Let μ be a finite Borel measure on S n − 1 , p ∈ R , and q ≥ 0 . What are the necessary and sufficient conditions for the existence of a convex body K ∈ K o n that solves the equation:

(1.2) F p , q ( K , ⋅ ) = μ ?

When p = 1 , it is the chord Minkowski problem, and the q = 1 case is the L p Minkowski problem.

When the given measure μ has a density f that is an integrable nonnegative function on S n − 1 , equation (1.2) becomes a new Monge-Ampère-type partial differential equation:

det ( ∇ i j h + h δ i j ) = h p − 1 f V ˜ q − 1 ( [ h ] , ∇ h ) , on S n − 1 ,

where h is the unknown function on S n − 1 , which is extended via homogeneity to R n , while ∇ h is the Euclidean gradient of h in R n , the spherical Hessian of h with respect to an orthonormal frame on S n − 1 is ( ∇ i j h ) , δ i j is the Kronecker delta, and V ˜ q − 1 ( [ h ] , ∇ h ) is the ( q − 1 ) th dual quermassintegral of the Wulff-shape [ h ] of h with respect to the point ∇ h (see next section for the precise definition).

We first solve the symmetric case of the L p chord Minkowski problem when p , q > 0 .

Theorem 1.1

Let p , q > 0 . If μ is an even finite Borel measure on S n − 1 that is not concentrated on a great subsphere, then there exists a symmetric, convex body K ∈ K e n such that

F p , q ( K , ⋅ ) = μ , when p ≠ n + q − 1 , F p , q ( K , ⋅ ) V ( K ) = μ , when p = n + q − 1 .

When q = 1 , this result is a solution to the symmetric L p Minkowski problem, see [18,36,42]. When p > 1 , the symmetric condition can be dropped. We have the following solution:

Theorem 1.2

Let p > 1 and q > 0 . If μ is a finite Borel measure on S n − 1 that is not concentrated in any closed hemisphere, then there exists a convex body K with nonnegative support function h K ≥ 0 so that

d F q ( K , ⋅ ) = h K p − 1 d μ , when p ≠ n + q − 1 , d F q ( K , ⋅ ) V ( K ) = h K p − 1 d μ , when p = n + q − 1 .

Moreover, h K > 0 if μ is discrete or if p ≥ n .

Again, when q = 1 , this result is a solution to the L p Minkowski problem, see [9,28].

2 Preliminaries

2.1 Wulff shape

Let Ω ⊂ S n − 1 be a closed subset that is not contained in any hemisphere. For h ∈ C ( Ω ) , the Wulff-shape [ h ] is a compact convex set defined by

[ h ] = { x ∈ R n : x ⋅ v ≤ h ( v ) , ∀ v ∈ Ω } .

Clearly, h [ h ] ( v ) ≤ h ( v ) . A useful fact is that, when [ h ] ∈ K n , the support of S n − 1 ( [ h ] , ⋅ ) must be contained in Ω .

In particular, let v 1 , … , v N ( N > n + 1 ) be unit vectors that are not contained in any closed hemisphere, and let Ω = { v 1 , … , v N } . For z = ( z 1 , … , z N ) ∈ R N , we write

[ z ] = P ( z ) = ⋂ k = 1 N { x ∈ R n : x ⋅ v i ≤ z i } .

Define P ( v 1 , … , v N ) by

P ( v 1 , … , v N ) = { P ( z ) : z ∈ R N such that P ( z ) ∈ K n } .

Denote R + = ( 0 , ∞ ) . Clearly, if z ∈ R + N , then P ( z ) ∈ K o n .

2.2 Chord integrals and chord measures

Let K ∈ K n . For z ∈ int K and q ∈ R , the q th dual quermassintegral V ˜ q ( K , z ) of K with respect to z is

V ˜ q ( K , z ) = 1 n ∫ S n − 1 ρ K , z ( u ) q d u ,

where ρ K , z ( u ) = max { λ > 0 : z + λ u ∈ K } is the radial function of K with respect to z . When z ∈ ∂ K , V ˜ q ( K , z ) is defined in the way that the integral is only over those u ∈ S n − 1 such that ρ K , z ( u ) > 0 . In another word,

V ˜ q ( K , z ) = 1 n ∫ ρ K , z ( u ) > 0 ρ K , z ( u ) q d u , whenever z ∈ ∂ K .

In this case, for ℋ n − 1 -almost all z ∈ ∂ K , we have

V ˜ q ( K , z ) = 1 2 n ∫ S n − 1 X K ( z , u ) q d u ,

where the parallel X-ray of K is the nonnegative function on R n × S n − 1 defined by

X K ( z , u ) = ∣ K ∩ ( z + R u ) ∣ , z ∈ R n , u ∈ S n − 1 .

When restricting to q > 0 , the dual quermassintegral is the Riesz potential of the characteristic function, that is,

V ˜ q ( K , z ) = q n ∫ K ∣ x − z ∣ q − n d x .

Note that this immediately allows an extension of V ˜ q ( K , ⋅ ) to R n . See [38] for an equivalent definition via radial function. By a change-of-variable, we have

V ˜ q ( K , z ) = q n ∫ K − z ∣ y ∣ q − n d y ,

and since, for q > 0 , the integrand ∣ y ∣ q − n is locally integrable, we immediately conclude that in this case, the dual quermassintegral V ˜ q ( K , z ) is continuous in z .

Let K ∈ K n . The X -ray X K ( x , u ) and the radial function ρ K , z ( u ) have the following relation:

(2.1) X K ( x , u ) = ρ K , z ( u ) + ρ K , z ( − u ) , when K ∩ ( x + R u ) = K ∩ ( z + R u ) ≠ ∅ .

When z ∈ ∂ K , then either ρ K , z ( u ) = 0 or ρ K , z ( − u ) = 0 for almost all u ∈ S n − 1 , and thus

X K ( z , u ) = ρ K , z ( u ) , or X K ( z , u ) = ρ K , z ( − u ) , z ∈ ∂ K ,

for almost all u ∈ S n − 1 . Then, the chord integral I q ( K ) can be represented as follows:

I q ( K ) = 1 n ω n ∫ S n − 1 ∫ u ⊥ X K ( x , u ) q d x d u , q ≥ 0 .

An elementary property of the functional I q is its homogeneity. If K ∈ K n and q ≥ 0 , then

I q ( t K ) = t n + q − 1 I q ( K ) ,

for t > 0 . By compactness of K , it is simple to see that the chord integral I q ( K ) is finite whenever q ≥ 0 .

Let K ∈ K n and q > 0 . The chord measure F q ( K , ⋅ ) is a finite Borel measure on S n − 1 given by

F q ( K , η ) = 2 q ω n ∫ ν K − 1 ( η ) V ˜ q − 1 ( K , z ) d ℋ n − 1 ( z ) , for each Borel η ⊂ S n − 1 ,

where ν K : ∂ K → S n − 1 is the Gauss map that takes boundary points of K to their corresponding outer unit normals. Note that by convexity of K , its Gauss map ν K is almost everywhere defined on ∂ K with respect to the ( n − 1 ) -dimensional Hausdorff measure. The significance of the chord measure F q ( K , ⋅ ) is that it comes from differentiating, in a certain sense, the chord integral I q , see (1.1). It is simple to see that the chord measure F q ( K , ⋅ ) is absolutely continuous with respect to the surface area measure S n − 1 ( K , ⋅ ) . In particular, for each P ∈ P ( v 1 , … , v N ) , we have that the chord measure F q ( P , ⋅ ) is supported entirely on { v 1 , … , v N } . It was shown in [38, Theorem 4.3] that

(2.2) I q ( K ) = 1 n + q − 1 ∫ S n − 1 h K ( v ) d F q ( K , v ) .

When q > 0 , a useful integral formula demonstrated in [38, Lemma 5.3] is

2 n ∫ ∂ K V ˜ q − 1 ( K , z ) g ( ν K ( z ) ) d ℋ n − 1 ( z ) = ∫ S n − 1 ∫ ∂ K X K ( z , u ) q − 1 g ( ν K ( z ) ) d ℋ n − 1 ( z ) d u ,

for any g ∈ C ( S n − 1 ) . Therefore, for each K ∈ K n , we have

(2.3) ∫ S n − 1 g ( v ) d F q ( K , v ) = q n ω n ∫ S n − 1 ∫ ∂ K X K ( z , u ) q − 1 g ( ν K ( z ) ) d ℋ n − 1 ( z ) d u = q n ω n ∫ S n − 1 ∫ S n − 1 X K ( ρ K ( w ) w , u ) q − 1 h K ( α K ( w ) ) − 1 ρ K ( w ) n g ( α K ( w ) ) d w d u .

Here, α K ( w ) = ν K ( w ρ K ( w ) ) is the radial Gauss map, and we have used the short-hand ρ K = ρ K , o .

For each p ∈ R and K ∈ K o n , the L p chord measure F p , q ( K , ⋅ ) is defined as follows:

d F p , q ( K , v ) = h K ( v ) 1 − p d F q ( K , v ) .

It was shown in [38] that the differential of the chord integral I q with respect to the L p Minkowski combinations leads to the L p chord measure: for p ≠ 0 ,

d d t t = 0 I q ( K + p t ⋅ L ) = 1 p ∫ S n − 1 h L p ( v ) d F p , q ( K , v ) ,

where K + p t ⋅ L is the L p Minkowski combination between K and L defined via Wulff shape as follows:

K + p t ⋅ L = ( h K p + t h L p ) 1 p .

It is worth pointing out that there is a similar formula for the p = 0 case, which leads to the cone-chord measure studied in [38]. Since the cone-chord measure and its Minkowski problem are not considered in the current work, we omit this formulation.

2.3 Weak continuity of L p chord measures

We prove in this subsection the weak continuity of the L p chord measure F p , q ( K , ⋅ ) in K with respect to the Hausdorff metric.

For each x ∈ R n and u ∈ S n − 1 , we will write x ∣ u ⊥ as the image point of the orthogonal projection of x onto u ⊥ . Similarly, for each subset E ⊂ R n , we write

E ∣ u ⊥ = { x ∣ u ⊥ : x ∈ E } .

We will need the following lemma obtained in [23].

Lemma 2.1

[23, Lemma 2.2] Let K i ∈ K o n be such that K i → K ∈ K o n in the Hausdorff metric as i → ∞ . Then, for ℋ n − 1 -a.e. w ∈ S n − 1 ,

α K i ( w ) → α K ( w ) , a s i → ∞ .

A generalized dominated convergence theorem will be needed to establish the weak continuity of chord measures: Suppose f k , ϕ k , f , and ϕ are integrable functions in a measure space with f k → f and ϕ k → ϕ , while ∣ f k ∣ ≤ ϕ k , almost everywhere. If ∫ ϕ k → ∫ ϕ , then ∫ f k → ∫ f .

We first show that chord measures are weakly continuous.

Theorem 2.2

Let q > 0 and K i ∈ K n . If K i → K ∈ K n , then the chord measure F q ( K i , ⋅ ) converges to F q ( K , ⋅ ) weakly.

Proof

Since the chord measure is translation invariant, we can assume without loss of generality that K i , K ∈ K o n . Then, by (2.3), for any g ∈ C ( S n − 1 ) ,

∫ S n − 1 g ( v ) d F q ( K i , v ) = q n ω n ∫ S n − 1 ∫ S n − 1 X K i ( ρ K i ( w ) w , u ) q − 1 h K i ( α K i ( w ) ) − 1 ρ K i ( w ) n g ( α K i ( w ) ) d w d u .

Let

f i ( w , u ) = X K i ( ρ K i ( w ) w , u ) q − 1 h K i ( α K i ( w ) ) − 1 ρ K i ( w ) n g ( α K i ( w ) )

and

f ( w , u ) = X K ( ρ K ( w ) w , u ) q − 1 h K ( α K ( w ) ) − 1 ρ K ( w ) n g ( α K ( w ) ) .

On the one hand, since K i → K ∈ K o n , we have ρ K i ( w ) → ρ K ( w ) . This and Lemma 2.1 further show that h K i ( α K i ( w ) ) → h K ( α K ( w ) ) a.e. w ∈ S n − 1 . Moreover, g ∈ C ( S n − 1 ) and Lemma 2.1 imply g ( α K i ( w ) ) → g ( α K ( w ) ) a.e. w ∈ S n − 1 . Note that ∂ K is line-free in direction u for almost all u ∈ S n − 1 . For such a u , the projection point ( ρ K ( w ) w ) ∣ u ⊥ belongs to int ( K ∣ u ⊥ ) (relative interior) for almost all w ∈ S n − 1 . Then,

X K i ( ρ K i ( w ) w , u ) → X K ( ρ K ( w ) w , u ) , a.e. w ∈ S n − 1 .

Overall, we have

f i ( w , u ) → f ( w , u ) , as i → ∞ ,

for ℋ n − 1 × ℋ n − 1 -a.e. ( w , u ) ∈ S n − 1 × S n − 1 .

On the other hand, since g ∈ C ( S n − 1 ) and K i → K ∈ K o n , there exists a constant c > 0 , such that

∣ g ( v ) ∣ ≤ c h K i ( v ) − 1 for all i ∈ N , v ∈ S n − 1 .

Let

ϕ i ( w , u ) = c X K i ( ρ K i ( w ) w , u ) q − 1 ρ K i ( w ) n

and

ϕ ( w , u ) = c X K ( ρ K ( w ) w , u ) q − 1 ρ K ( w ) n .

Then,

∣ f i ∣ ≤ ϕ i and ∣ f ∣ ≤ ϕ .

By (2.2), we have

∫ S n − 1 ∫ S n − 1 ϕ i ( w , u ) d w d u = c n ( n + q − 1 ) ω n q I q ( K i )

and

∫ S n − 1 ∫ S n − 1 ϕ ( w , u ) d w d u = c n ( n + q − 1 ) ω n q I q ( K ) .

By the fact that I q ( K i ) → I q ( K ) , applying the generalized dominated convergence theorem, we obtain

lim i → ∞ ∫ S n − 1 ∫ S n − 1 f i ( w , u ) d w d u = ∫ S n − 1 ∫ S n − 1 f ( w , u ) d w d u ,

which completes the proof.□

As a corollary, one immediately obtains the following weak continuity property for L p chord measures.

Corollary 2.3

Let q > 0 and K i ∈ K n . If K i → K ∈ K n ,

when p < 1 , with the additional assumption that o ∈ K i ∩ K , then F p , q ( K i , ⋅ ) converges to F p , q ( K , ⋅ ) weakly;
when p > 1 , with the additional assumption that o ∈ int ( K i ) ∩ int ( K ) , then F p , q ( K i , ⋅ ) converges to F p , q ( K , ⋅ ) weakly.

Proof

Note that

d F p , q ( K i , ⋅ ) = h K i 1 − p d F q ( K i , ⋅ ) .

In the case p < 1 , note that since both h K i and h K are nonnegative, we obtain from the uniform convergence of h K i to h K the fact that h K i 1 − p converges to h K 1 − p uniformly. In the case p > 1 , with the additional assumption that both K i and K contain the origin in their respective interiors, along with the fact that K i → K , we conclude that h K i and h K are uniformly bounded away from 0. As a consequence, we also have the uniform convergence of h K i 1 − p to h K 1 − p .

The desired weak convergence now readily follows from Theorem 2.2.□

3 Variation problems for chord measures

3.1 Variation formula

The following variational formula was shown in [38].

Theorem 3.1

[38, Theorem 5.5] Let q > 0 , and Ω be a compact subset of S n − 1 that is not contained in any closed hemisphere. Suppose that g : Ω → R is continuous and h t : Ω → ( 0 , ∞ ) is a family of continuous functions given as follows:

h t = h 0 + t g + o ( t , ⋅ ) ,

for each t ∈ ( − δ , δ ) for some δ > 0 . Here, o ( t , ⋅ ) ∈ C ( Ω ) and o ( t , ⋅ ) / t tends to 0 uniformly on Ω as t → 0 . Let K t be the Wulff shape generated by h t and K be the Wulff shape generated by h 0 . Then,

(3.1) d d t t = 0 I q ( K t ) = ∫ Ω g ( v ) d F q ( K , v ) .

Remark 3.2

Note that the aforementioned theorem is slightly different from Theorem 5.5 in [38]. Indeed, the domain of g in Theorem 5.5 in [38] is S n − 1 and is changed to Ω here. Despite the change, the proof, however, works for any Ω without any essential changes once we realize the fact that, for h : Ω → ( 0 , ∞ ) and for almost all x ∈ ∂ [ h ] , we have ν [ h ] ( x ) ∈ Ω . For completeness, we include a detailed proof in the Appendix.

Note that the special case of q = 1 of the variational formula (3.1) is the volume variational formula of Aleksandrov. Taking Ω to be a finite set, we immediately obtain the following corollary for the discrete case.

Corollary 3.3

Let p , q > 0 , z = ( z 1 , … , z N ) ∈ R + N , β = ( β 1 , … , β N ) ∈ R N , and v 1 , … , v N be N unit vectors that are not contained in any closed hemisphere. For sufficiently small ∣ t ∣ , consider z i ( t ) p = z i p + t β i > 0 and

P t = [ z ( t ) ] = ⋂ i = 1 N { x ∈ R n : x ⋅ v i ≤ z i ( t ) = ( z i p + t β i ) 1 p } .

Then, for q > 0 , we have

(3.2) d d t t = 0 I q ( P t ) = 1 p ∑ i = 1 N β i F p , q ( P 0 , v i ) .

Here, in proving (3.2), we used the fact that F q ( P 0 , ⋅ ) is supported entirely on { v 1 , … , v N } .

3.2 Maximization problems

The goal of this subsection is to convert the existence of the solution to the L p chord Minkowski problem to the existence of the solution to maximization problem.

For each Ω ⊂ S n − 1 , we will write C ( Ω ) for the set of continuous functions on Ω . The set C + ( Ω ) ⊂ C ( Ω ) will denote the subset consisting only of positive functions. Similarly, the set C e + ( Ω ) consists only of positive, symmetric continuous functions on Ω . We will write supp μ for the support of a measure μ .

Let p , q ≠ 0 and Ω ⊂ S n − 1 be a compact subset that is not contained in any closed hemisphere. For each non-zero finite Borel measure μ on S n − 1 , define the functional Φ p , q : C + ( Ω ) → R n by

Φ p , q ( h ) = 1 n + q − 1 log I q ( [ h ] ) − 1 p log ∫ Ω h ( v ) p d μ ( v ) .

Theorem 3.4

Let p > 1 , q > 0 , and μ be a nonzero finite Borel measure on S n − 1 that is not concentrated in any closed hemisphere. Suppose Ω ⊂ S n − 1 is a compact subset such that supp μ ⊂ Ω . If the maximization problem

sup { Φ p , q ( h ) : h ∈ C + ( Ω ) }

has a solution h 0 ∈ C + ( Ω ) , then there exists K 0 ∈ K o n such that

F p , q ( K 0 , ⋅ ) = μ , if p ≠ n + q − 1 , F p , q ( K 0 , ⋅ ) V ( K 0 ) = μ , if p = n + q − 1 .

Proof

Let g ∈ C ( Ω ) . Define h t = h 0 + t g . For sufficiently small ∣ t ∣ , the family h t ∈ C + ( Ω ) . Using the fact that h 0 is a maximizer and Theorem 3.1, we have

0 = 1 n + q − 1 d d t t = 0 ( log I q ( [ h t ] ) ) − 1 p d d t t = 0 log ∫ Ω h t ( v ) p d μ ( v ) = 1 ( n + q − 1 ) I q ( [ h 0 ] ) ∫ Ω g ( v ) d F q ( [ h 0 ] , v ) − ∫ Ω g ( v ) h 0 p − 1 ( v ) d μ ( v ) ∫ Ω h 0 p d μ .

Since g ∈ C ( Ω ) is arbitrary and using the fact that h 0 = h [ h 0 ] , F q ( [ h 0 ] , ⋅ ) -almost everywhere, we have

(3.3) F p , q ( [ h 0 ] , ⋅ ) I q ( [ h 0 ] ) = ( n + q − 1 ) ∫ Ω h 0 p d μ μ ( ⋅ ) on Ω .

Note that the measure F p , q ( K , ⋅ ) is homogeneous of degree n + q − p − 1 in K . Therefore, we may rescale [ h 0 ] and obtain K 0 ∈ K o n such that

F p , q ( K 0 , ⋅ ) = μ on Ω

if p ≠ n + q − 1 , and

F p , q ( K 0 , ⋅ ) V ( K 0 ) = μ on Ω

if p = n + q − 1 . Noting that both μ and F p , q ( K 0 , ⋅ ) are concentrated on Ω , we reach the desired conclusion.□

Remark 3.5

In fact, it is clear from the proof that the convex body K 0 obtained in Theorem 3.4 is a rescaling of [ h 0 ] , that is K 0 = c [ h 0 ] , where

c = ∫ Ω h 0 p d μ ( n + q − 1 ) I q ( [ h 0 ] ) 1 n + q − p − 1 ,

if p ≠ n + q − 1 . If p = n + q − 1 ,

c = I 1 ( [ h 0 ] ) ∫ Ω h 0 p d μ ( n + q − 1 ) I q ( [ h 0 ] ) − 1 n .

Taking Ω = { v 1 , … , v N } , where the v i ∈ S n − 1 are not contained entirely in any closed hemisphere, from the fact that Φ p , q is homogeneous of degree 0, we immediately obtain the following discrete version of the maximization problem.

Theorem 3.6

Let p > 1 , q > 0 , and

μ = ∑ i = 1 N α i δ { v i }

be a finite discrete measure on S n − 1 , where α i > 0 and v i ∈ S n − 1 . Suppose v 1 , … , v N are not contained entirely in any closed hemisphere. If the maximization problem

(3.4) sup I q ( P ( z ) ) : ∑ i = 1 N α i z i p ≤ 1 , z = ( z 1 , … , z N ) ∈ R + N

has a solution z 0 ∈ R + N , then there exists a polytope P 0 containing the origin in its interior such that

F p , q ( P 0 , ⋅ ) = μ , if p ≠ n + q − 1 , F p , q ( P 0 , ⋅ ) V ( P 0 ) = μ , if p = n + q − 1 .

Using virtually the same argument, we may obtain the symmetric version of Theorem 3.4.

Theorem 3.7

Let p , q > 0 and μ be a nonzero even finite Borel measure on S n − 1 that is not concentrated in any closed hemisphere. If the maximization problem

(3.5) sup { Φ p , q ( h ) : h ∈ C e + ( S n − 1 ) }

has a solution h 0 ∈ C e + ( S n − 1 ) , then there exists K 0 ∈ K e n such that

F p , q ( K 0 , ⋅ ) = μ , if p ≠ n + q − 1 , F p , q ( K 0 , ⋅ ) V ( K 0 ) = μ , if p = n + q − 1 .

4 The even L p chord Minkowski problem when p , q > 0

In this section, we solve the even L p chord Minkowski problem when p , q > 0 .

We will use B to denote the centered unit ball in R n and we will write ( ⋅ ) + : R → [ 0 , ∞ ) for the function given as

( t ) + = t , if t ≥ 0 , 0 , otherwise ,

for each t ∈ R .

Lemma 4.1

Let p > 0 and μ be a finite Borel measure that is not concentrated in any closed hemisphere. If K ∈ K o n and

∫ S n − 1 h K ( v ) p d μ ( v ) ≤ 1 ,

then there exists a positive constant c p ( μ ) depending only on μ and p, such that

K ⊂ c p ( μ ) B ,

In particular,

(4.1) c p ( μ ) − p = min u ∈ S n − 1 ∫ S n − 1 ( u ⋅ v ) + p d μ ( v ) > 0 .

Proof

Since μ is not concentrated in any closed hemisphere, the function

u ↦ ∫ S n − 1 ( u ⋅ v ) + p d μ ( v )

is strictly positive on S n − 1 and continuous. By the compactness of S n − 1 , the constant c p ( μ ) − p defined in (4.1) is positive.

Denote R 0 = max { ∣ x ∣ : x ∈ K } . Then, there exists u 0 ∈ S n − 1 such that R 0 u 0 ∈ K . Note that o ∈ K . Thus, we obtain

R 0 p ⋅ c p ( μ ) − p ≤ ∫ S n − 1 ( R 0 u 0 ⋅ v ) + p d μ ( v ) ≤ ∫ S n − 1 h K ( v ) p d μ ( v ) ≤ 1 ,

and hence R 0 ≤ c p ( μ ) , which implies what we wanted by the choice of R 0 .□

The following theorem is Theorem 1.1.

Theorem 4.2

Let p , q > 0 . If μ is an even finite Borel measure on S n − 1 that is not concentrated on a great subsphere, then there exists a symmetric, convex body K ∈ K e n such that

(4.2) F p , q ( K , ⋅ ) = μ , when p ≠ n + q − 1 ,

(4.3) F p , q ( K , ⋅ ) V ( K ) = μ , when p = n + q − 1 .

Proof

Let h i be a sequence of functions in C e + ( S n − 1 ) such that

Φ p , q ( h i ) → sup { Φ p , q ( h ) : h ∈ C e + ( S n − 1 ) } .

Denote K i = [ h i ] . Since Φ p , q ( ⋅ ) is 0-homogeneous, we may assume that

∫ S n − 1 h i p d μ = 1 .

It follows from Lemma 4.1 and the fact h K i ≤ h i that K i are uniformly bounded. By Blaschke’s selection theorem, there exists a subsequence, which will still be denoted as K i (since there is no confusion), such that

K i → K 0 ,

where K 0 is origin-symmetric, convex, and compact.

If int K 0 is empty, then I q ( K i ) → 0 , and hence Φ p , q ( h i ) → − ∞ , which contradicts the fact that h i is a maximizing sequence. Thus, K 0 ∈ K e n , and as a result, h 0 = h K 0 ∈ C e + ( S n − 1 ) . Moreover,

Φ p , q ( h 0 ) = 1 n + q − 1 log I q ( [ h 0 ] ) − 1 p log ∫ S n − 1 h 0 ( v ) p d μ ( v ) = lim i → ∞ 1 n + q − 1 log I q ( [ h i ] ) − 1 p log ∫ S n − 1 h K i ( v ) p d μ ( v ) ≥ lim i → ∞ 1 n + q − 1 log I q ( [ h i ] ) − 1 p log ∫ S n − 1 h i ( v ) p d μ ( v ) = lim i → ∞ Φ p , q ( h i ) .

Therefore, h 0 is a maximizer for the maximization problem (3.5). By Theorem 3.7, we obtain (4.2) and (4.3).□

5 The discrete L p chord Minkowski problem when p > 1 and q > 0

In this section, we solve the discrete L p chord Minkowski problem when p > 1 and q > 0 without the assumption that the given measure μ is even.

The following lemma shows the non-degeneracy of the maximizer to the maximization problem (3.4).

Lemma 5.1

Let p > 1 , q > 0 , and

μ = ∑ i = 1 N α i δ { v i }

be a finite discrete measure on S n − 1 , where α i > 0 and v i ∈ S n − 1 . Suppose v 1 , … , v N are not contained entirely in any closed hemisphere. If z 0 = ( z 1 0 , … z N 0 ) ∈ R N is such that z i 0 ≥ 0 and satisfies

∑ i = 1 N α i ( z i 0 ) p ≤ 1

and

(5.1) I q ( P ( z 0 ) ) = sup I q ( P ( z ) ) : ∑ i = 1 N α i z i p ≤ 1 , z = ( z 1 , … , z N ) ∈ R + N ,

then z 0 ∈ R + N .

Proof

We argue by contradiction and assume that at least one z i 0 = 0 . We write P 0 = P ( z 0 ) . Therefore, o ∈ ∂ P 0 . For simplicity, we write h i = h P 0 ( v i ) ≥ 0 . Let

J = { i : h i = 0 } .

Since o ∈ ∂ P 0 , we have that J is non-empty. It is also simple to see that J ≠ { 1 , … , N } . Indeed, if that is not the case, then P 0 = { o } , which implies that I q ( P ( z 0 ) ) = 0 and this is a contradiction to (5.1). (Clearly, there is some z ∈ R + N in the domain of the maximization problem (5.1) and I q ( P ( z ) ) > 0 .) By the same argument, P 0 must have a nonempty interior.

For each t > 0 , let

z i ( t ) = t 1 p , if i ∈ J , ( h i p − a t ) 1 p , if i ∉ J ,

where

a = ∑ i ∈ J α i ∑ i ∉ J α j > 0 .

Clearly, for sufficiently small t > 0 , we have z ( t ) = ( z 1 ( t ) , … , z N ( t ) ) ∈ R + N . Note that

∑ i = 1 N α i z i ( t ) p = ∑ i ∈ J α i t + ∑ i ∉ J α i ( h i p − a t ) = ∑ i ∉ J α i h i p = ∑ i = 1 N α i h i p ≤ ∑ i = 1 N α i ( z i 0 ) p ≤ 1 .

For simplicity, write P t = P ( z ( t ) ) . Note that according to Corollary 3.3, the functional I q ( P t ) is differentiable in t for sufficiently small t > 0 . Note also that I q ( P t ) is continuous at t = 0 . Therefore, by the mean value theorem and the fact that p > 1 ,

p ⋅ I q ( P t ) − I q ( P 0 ) t = p ⋅ d d t t = θ I q ( P t ) = ∑ i ∈ J h P θ ( v i ) 1 − p F q ( P θ , v i ) − ∑ i ∉ J a h P θ ( v i ) 1 − p F q ( P θ , v i ) ≥ ∑ i ∈ J t 1 − p p F q ( P θ , v i ) − ∑ i ∉ J a h P θ ( v i ) 1 − p F q ( P θ , v i )

for some θ ∈ ( 0 , t ) that depends on t .

Since o ∈ ∂ P 0 , there must exist i 0 ∈ J such that P 0 has a facet (with positive ℋ n − 1 -area) with v i 0 as its normal. Therefore, by the definition of F q , we have F q ( P 0 , { v i 0 } ) > 0 . By Theorem 2.2, when t > 0 is sufficiently small, we have

F q ( P θ , v i 0 ) ≥ 1 2 F q ( P 0 , v i 0 ) > 0 .

By the fact that P t → P 0 and the choice of J , we know that for t > 0 sufficiently small, there exists c 0 > 0 such that h P θ ( v i ) ≥ c 0 for each i ∉ J . Combining these facts together with (2.2), we have

p ⋅ I q ( P t ) − I q ( P 0 ) t ≥ t 1 − p p 1 2 F q ( P 0 , v i 0 ) − a c 0 − p ∑ i ∉ J h P θ ( v i ) F q ( P θ , v i ) ≥ t 1 − p p 1 2 F q ( P 0 , v i 0 ) − a c 0 − p ∑ i = 1 N h P θ ( v i ) F q ( P θ , v i ) = t 1 − p p 1 2 F q ( P 0 , v i 0 ) − a c 0 − p ( n + q − 1 ) I q ( P θ ) > 0 ,

when t > 0 is sufficiently small. This implies the existence of t 0 > 0 such that I q ( P t 0 ) > I q ( P 0 ) , or, equivalently, I q ( z ( t 0 ) ) > I q ( P ( z 0 ) ) . This is in contradiction to (5.1).□

With the above lemma, we obtain the following solution to the discrete L p chord Minkowski problem when p > 1 , q > 0 .

Theorem 5.2

Let p > 1 , q > 0 , and μ be a discrete measure on S n − 1 that is not concentrated in any closed hemisphere. Then, there is a polytope P ∈ K o n such that

F p , q ( P , ⋅ ) = μ , when p ≠ n + q − 1 ,

F p , q ( P , ⋅ ) V ( P ) = μ , when p = n + q − 1 .

Proof

Suppose

μ = ∑ i = 1 N α i δ { v i } ,

where v 1 , … , v N are N unit vectors not contained in any closed hemisphere, and α 1 , … , α N > 0 .

Let z ( k ) = ( z 1 ( k ) , … , z N ( k ) ) ∈ R + N be a maximizing sequence to (3.4), that is,

(5.2) ∑ i = 1 N α i z i ( k ) p ≤ 1

and

I q ( P ( z ( k ) ) ) → sup I q ( P ( z ) ) : ∑ i = 1 N α i z i p ≤ 1 , z = ( z 1 , … , z n ) ∈ R + N .

By (5.2) and the fact that α i > 0 , it is clear that z i ( k ) are uniformly bounded in k and i . Therefore, we may pick a subsequence, which we still denote by z ( k ) such that z ( k ) → z 0 ∈ R N . Since z i ( k ) > 0 , we have z i 0 ≥ 0 . Moreover, we have

∑ i = 1 N α i ( z i 0 ) p = lim k → ∞ ∑ i = 1 N α i z i ( k ) p ≤ 1

and

I q ( P ( z 0 ) ) = lim k → ∞ I q ( P ( z ( k ) ) ) = sup I q ( P ( z ) ) : ∑ i = 1 N α i z i p ≤ 1 , z = ( z 1 , … , z n ) ∈ R + N .

Therefore, by Lemma 5.1, we conclude that z 0 ∈ R + N . In fact, since z 0 maximizes (3.4), by the monotonicity of I q , it must be the case that

∑ i = 1 N α i ( z i 0 ) p = 1 .

Therefore, z 0 is a maximizer to the maximization problem (3.4). Now, we may use Theorem 3.6 to find the desired polytope P ∈ K o n .□

A normalized version of Theorem 5.2 together with a boundness estimate will be useful in the next section.

Corollary 5.3

Let p , q , and μ be the same as in Theorem 5.2. Then, there is a polytope P 0 such that

(5.3) F p , q ( P 0 , ⋅ ) I q ( P 0 ) = μ ( ⋅ ) ,

and P 0 satisfies

(5.4) P 0 ⊂ ( n + q − 1 ) 1 p c p ( μ ) B ,

where c p ( μ ) is given in Lemma 4.1.

Proof

Following the proof of Theorem 5.2, we may find a maximizer z to the maximization problem (3.4), and z satisfies

∑ i = 1 N α i z i p = 1 .

From (3.3) in the proof of Theorem 3.4, we have that P 0 = ( n + q − 1 ) 1 p ⋅ [ z ] satisfies (5.3). The desired bound (5.4) for P 0 follows immediately from Lemma 4.1.□

6 The L p chord Minkowski problem for general measures when p > 1 and q > 0

This section is focused on solving the existence of a solution to the L p chord Minkowski problem when p > 1 , q > 0 , and the given measure μ is not necessarily discrete or even.

Lemma 6.1

Let p ≥ 1 and ( μ i ) i ∈ N be a sequence of nonzero finite Borel measures that converges weakly to a nonzero finite Borel measure μ . Suppose μ is not concentrated entirely in any closed hemisphere. Then, for sufficiently large i, we have

c p ( μ i ) ≤ 2 c p ( μ ) .

Here, c p ( μ ) is defined by (4.1) in Lemma 4.1.

Proof

Define

f μ ( u ) = ∫ S n − 1 ( u ⋅ v ) + p d μ ( v ) .

Since μ i ⇀ μ weakly,

f μ i ( u ) → f μ ( u ) , ∀ u ∈ S n − 1 .

Since f μ i 1 / p and f μ 1 / p are support functions, point-wise convergence implies uniform convergence. Note that f μ is always positive following the fact that μ is not concentrated in any closed hemisphere. Therefore, when i is sufficiently large,

c p ( μ i ) = ( min u ∈ S n − 1 f μ i ( u ) ) − 1 / p ≤ 2 ( min u ∈ S n − 1 f μ ( u ) ) − 1 / p = 2 c p ( μ ) . □

Theorem 6.2

d F q ( K , ⋅ ) = h K p − 1 d μ , when p ≠ n + q − 1 , d F q ( K , ⋅ ) V ( K ) = h K p − 1 d μ , when p = n + q − 1 .

Proof

Choose a sequence of discrete measure ( μ i ) i ∈ N such that μ i converges to μ weakly. Since μ is not concentrated in any closed hemisphere, we may choose μ i so that μ i is not concentrated in any closed hemisphere either.

By Corollary 5.3, for each i , there is a polytope P i ∈ K o n such that

I q ( P i ) h P i ( v ) p − 1 d μ i ( v ) = d F q ( P i , v )

and

P i ⊂ ( n + q − 1 ) 1 p c p ( μ i ) B .

By Lemma 6.1, ( P i ) i ∈ N is uniformly bounded. Thus, there is a subsequence of ( P i ) converging to a compact, convex set K 0 that contains the origin (not necessarily as an interior point). We claim that K 0 has nonempty interior. If this is not the case, then I ( P i ) → 0 . This is in contradiction to P i being (rescaled versions of) the maximizer to (5.1) (with μ replaced by μ i ) and the fact that μ i converges to μ weakly.

By Theorem 2.2, the uniform convergence of support functions, and the continuity of the chord integral, we have

I q ( K 0 ) h K 0 ( v ) p − 1 d μ ( v ) = d F q ( K 0 , v ) .

By the homogeneities of F q ( K , ⋅ ) , I q ( K ) , V ( K ) , and h K in K , we may rescale K 0 and obtain K ∈ K n with nonnegative support function such that

d F q ( K , ⋅ ) = h K p − 1 d μ

if p ≠ n + q − 1 , and

d F q ( K , ⋅ ) V ( K ) = h K p − 1 d μ

if p = n + q − 1 .□

When p ≥ n , we may further show that the solution obtained in Theorem 6.2 contains the origin as an interior point. We require the following lemma.

Lemma 6.3

Let q > − 1 . If K i ∈ K n converges to K ∈ K n as i → ∞ , then there exists a positive constant c ( n , q , K ) such that for ℋ n − 1 -almost all z ∈ ∂ K i and every i ,

V ˜ q ( K i , z ) ≥ c ( n , q , K ) > 0 .

Proof

Recall that by convexity, for ℋ n − 1 -almost all z ∈ ∂ K i , there is a unique tangent plane to K i at z . As a consequence, the set

ℋ n − 1 ( { u ∈ S n − 1 : ρ K i , z ( u ) > 0 } ) = ω n 2 ,

for ℋ n − 1 -almost all z ∈ ∂ K i .

If q = 0 , based on the earlier observation, there is nothing to prove as V ˜ q ( K i , z ) is a constant ℋ n − 1 -almost everywhere.

If − 1 < q < 0 , then for ℋ n − 1 -almost all z ∈ ∂ K i ,

V ˜ q ( K i , z ) = 1 n ∫ S n − 1 ρ K i , z ( u ) q d u ≥ ω n 2 n D ( K i ) q > 0 ,

where D ( K i ) denotes the diameter of K i . Since K i → K , one has D ( K i ) → D ( K ) , and one can easily obtain the positive constant c ( n , q , K ) in this case.

Let us now deal with the q > 0 case. Recall that in this case V ˜ q ( K , z ) is continuous in z . Moreover, if L is a convex body contained in K ,

V ˜ q ( L , z ) ≤ V ˜ q ( K , z ) .

Since K i → K ∈ K n , there exist two balls B 1 and B 2 (not necessarily centered at the origin) such that

B 1 ⊂ K i ⊂ B 2 , ∀ i ∈ N .

Note that

V ˜ q ( K i , z ) ≥ V ˜ q ( B 1 , z ) , ∀ z ∈ ∂ K i .

Since ∂ K i ⊂ B 2 , we have

V ˜ q ( K i , z ) ≥ min z ∈ B 2 V ˜ q ( B 1 , z ) > 0 . □

The following lemma is extracted from Section 4 of [28].

Lemma 6.4

[28] Let p ≥ n . Suppose P i ∈ K o n are polytopes, and P i → K ∈ K n as i → ∞ . If there exists a constant c > 0 independent of i such that

∫ S n − 1 h P i 1 − p ( v ) d S P i ( v ) < c ,

then K contains the origin in its interior.

Theorem 6.5

If we further assume p ≥ n , the convex body K obtained in Theorem 6.2 must be in K o n . In particular, we have

(6.1) F p , q ( K , ⋅ ) = μ , when p ≠ n + q − 1 ,

(6.2) F p , q ( K , ⋅ ) V ( K ) = μ , when p = n + q − 1 .

Proof

Let P i be the convergent subsequence with limit K 0 that was obtained in the proof of Theorem 6.2.

By the definition of L p chord measure,

F p , q ( P i , S n − 1 ) = 2 q ω n ∫ ∂ P i V ˜ q − 1 ( P i , z ) ( z ⋅ ν K ( z ) ) 1 − p d ℋ n − 1 ( z ) .

By Lemma 6.3, there exists a positive uniform lower bound c ( n , q − 1 , K 0 ) of V ˜ q − 1 ( P i , z ) . Thus,

∫ S n − 1 h P i 1 − p ( v ) d S P i ( v ) ≤ ω n 2 q c ( n , q − 1 , K 0 ) F p , q ( P i , S n − 1 ) .

Since

F p , q ( P i , ⋅ ) I q ( P i ) = μ i ⇀ μ weakly ,

and I q ( P i ) → I q ( K 0 ) > 0 , we infer that

∫ S n − 1 h P i 1 − p ( v ) d S P i ( v )

has a uniform upper bound. It follows from Lemma 6.4 that K 0 ∈ K o n . Now (6.1) and (6.2) follow from this and Theorem 6.2.□

Theorem 1.2 follows from Theorems 5.2, 6.2, and 6.5.

Funding information: D.X. was supported, in part, by National Natural Science Foundation of China Grant 12071277 and Science and Technology Commission of Shanghai Municipality Grant 20JC1412600. D.Y. and G.Z. were supported, in part, by National Science Foundation (NSF) Grant DMS-2005875. Y.Z. was supported, in part, by NSF Grant DMS-2132330.
Conflict of interest: Prof. Gaoyong Zhang, who is the co-author of this article, is a current Editorial Board member of Advanced Nonlinear Studies. This fact did not affect the peer-review process. The authors declare no other conflict of interest.

Appendix

The aim of this appendix is to give detailed proof of Theorem 3.1, which is a general version of Theorem 5.5 in [38]. Its proof is actually based on a careful examination of the approach in [38].

Let Ω ⊂ S n − 1 be a compact set that is not contained in any closed hemisphere, g ∈ C ( Ω ) , and δ > 0 . Let h t : Ω → ( 0 , ∞ ) be a family of continuous functions defined by

h t ( v ) = h 0 ( v ) + t g ( v ) + o ( t , v ) , t ∈ ( − δ , δ ) and v ∈ Ω ,

where o ( t , ⋅ ) ∈ C ( Ω ) , and o ( t , ⋅ ) / t → 0 uniformly on Ω , as t → 0 . Recall that the Wulff-shape [ h t ] of h t is

(A1) [ h t ] = { x ∈ R n : x ⋅ v ≤ h t ( v ) for all v ∈ Ω } .

We require that [ h 0 ] has a nonempty interior throughout this section and as a consequence, the set [ h t ] also has nonempty interior for sufficiently small ∣ t ∣ .

The following differential formula was established in [23], for almost all u ∈ S n − 1 ,

(A2) d ρ [ h t ] ( u ) d t t = 0 = g ( ν [ h 0 ] ( y ) ) u ⋅ ν [ h 0 ] ( y ) ,

where y = ρ [ h 0 ] ( u ) u . We remark that even though g is only defined on Ω , the right side of (A2) makes sense for almost all u ∈ S n − 1 . This is because the normal vector ν [ h 0 ] ( x ) ∈ Ω for ℋ n − 1 -almost all x ∈ ∂ [ h 0 ] .

The following is the differential formula for the extended radial function, which is a slight extension of (A2). For a point z ∈ R n , we denote h t , z to be the translation of h t ,

h t , z ( v ) = h t ( v ) − z ⋅ v .

Lemma A.1

Let K t = [ h t ] be the Wulff shape defined by (A1) and K = K 0 . If z is an interior point of K, then for almost all u ∈ S n − 1 ,

(A3) d ρ K t , z ( u ) d t t = 0 = g ( ν K ( z + ρ K , z ( u ) u ) ) u ⋅ ν K ( z + ρ K , z ( u ) u ) .

Proof

Since

ρ K t , z ( u ) = ρ K t − z ( u ) ,

we obtain

K t − z = { x − z : x ∈ R n and x ⋅ v ≤ h t ( v ) , for all v ∈ Ω } = { y ∈ R n : y ⋅ v ≤ h 0 ( v ) − z ⋅ v + t g ( v ) + o ( t , v ) , for all v ∈ Ω } = [ h t , z ] .

Thus,

d ρ K t , z ( u ) d t t = 0 = d ρ h t , z ( u ) d t t = 0 .

Since z is an interior point of [ h 0 ] , the body [ h 0 , z ] = K − z contains the origin in its interior. By (A2),

d ρ [ h t , z ] t ( u ) d t t = 0 = g ( ν K ( y ) ) u ⋅ ν K ( y ) ,

where y = z + ρ K , z ( u ) u ∈ ∂ K . The desired formula (A3) follows.□

By using (A3), we now derive the differential formula for the X -ray function.

Lemma A.2

Let K t = [ h t ] be the Wulff shape defined by (A1) and K = K 0 . If u ∈ S n − 1 , then for almost all x in the interior of K ∣ u ⊥ ,

(A4) d X K t ( x , u ) d t t = 0 = g ( ν K ( y ) ) u ⋅ ν K ( y ) − g ( ν K ( y − ) ) u ⋅ ν K ( y − ) ,

where y and y − are the upper and lower points of ∂ K ∩ ( x + R u ) .

Proof

Since x is an interior point of K ∣ u ⊥ , we can pick an interior point z in K so that

K ∩ ( x + R u ) = K ∩ ( z + R u ) .

By (2.1) and (A3), we have

d X K t ( x , u ) d t t = 0 = d ρ K t , z ( u ) d t t = 0 + d ρ K t , z ( − u ) d t t = 0 = g ( ν K ( y ) ) u ⋅ ν K ( y ) − g ( ν K ( y − ) ) u ⋅ ν K ( y − ) . □

The following two lemmas from [38] are required.

Lemma A.3

[38, Lemma 5.3] Suppose K ∈ K n , q > − 1 , and g is a continuous function on S n − 1 . Then,

2 n ∫ ∂ K V ˜ q ( K , z ) g ( ν K ( z ) ) d ℋ n − 1 ( z ) = ∫ S n − 1 ∫ ∂ K X K ( z , u ) q g ( ν K ( z ) ) d ℋ n − 1 ( z ) d u .

Lemma A.4

[38, Lemma 4.8] If K ∈ K o n , then for all q > 0 ,

I q ( K ) = 2 q ( n + q − 1 ) n ω n ∫ ∂ K ∫ S z + ( z ⋅ ν K ( z ) ) ρ K , z ( u ) q − 1 d u d ℋ n − 1 ( z ) = 2 q ( n + q − 1 ) ω n ∫ ∂ K ( z ⋅ ν K ( z ) ) V ˜ q − 1 ( K , z ) d ℋ n − 1 ( z ) .

A generalized dominated convergence theorem will be needed to establish the lemma to follow: Suppose f k , ϕ k , f , and ϕ are integrable functions in a measure space with f k → f and ϕ k → ϕ , while ∣ f k ∣ ≤ ϕ k , almost everywhere. If ∫ ϕ k → ∫ ϕ , then ∫ f k → ∫ f . The following lemma is the crucial technical lemma needed in order to establish the differential formula for chord integrals.

Lemma A.5

Suppose q > 0 . Let K t = [ h t ] be the Wulff shape defined by (A1) and K = K 0 . Then, there is a class of nonnegative integrable functions ϕ t ( x , u ) defined for u ∈ S n − 1 and x ∈ u ⊥ such that

(A5) 1 t ( X K t ( x , u ) q − X K ( x , u ) q ) ≤ ϕ t ( x , u ) .

Moreover, the limit function lim t → 0 ϕ t ( x , u ) is integrable and

(A6) lim t → 0 ∫ S n − 1 ∫ u ⊥ ϕ t ( x , u ) d x d u = ∫ S n − 1 ∫ u ⊥ lim t → 0 ϕ t ( x , u ) d x d u .

Proof

Since h 0 is positive and continuous, the origin is inside the interior of K = [ h 0 ] . Since g in (A1) is continuous, and since o ( t , ⋅ ) / t → 0 uniformly on S n − 1 , there exist constants c , δ ′ > 0 so that

g ( v ) + o ( t , v ) t ≤ c h 0 ( v ) , for all v ∈ Ω , t ∈ ( − δ ′ , δ ′ ) .

Then,

( 1 − c ∣ t ∣ ) K ⊂ K t ⊂ ( 1 + c ∣ t ∣ ) K .

Thus,

(A7) 1 t ( X K t ( x , u ) q − X K ( x , u ) q ) ≤ ϕ t ( x , u ) ,

where

ϕ t ( x , u ) = 1 ∣ t ∣ ( X ( 1 + c ∣ t ∣ ) K ( x , u ) q − X ( 1 − c ∣ t ∣ ) K ( x , u ) q ) .

Thus, (A5) holds. The homogeneity of I q ( K , u ) , gives us

∫ u ⊥ ϕ t ( x , u ) d x = 1 ∣ t ∣ ( I q ( ( 1 + c ∣ t ∣ ) K , u ) − I q ( ( 1 − c ∣ t ∣ ) K , u ) ) = 1 ∣ t ∣ ( ( 1 + c ∣ t ∣ ) n + q − 1 − ( 1 − c ∣ t ∣ ) n + q − 1 ) I q ( K , u ) .

Therefore,

∫ S n − 1 ∫ u ⊥ ϕ t ( x , u ) d x d u = 1 ∣ t ∣ ( ( 1 + c ∣ t ∣ ) n + q − 1 − ( 1 − c ∣ t ∣ ) n + q − 1 ) n ω n I q ( K ) ,

and thus,

lim t → 0 ∫ S n − 1 ∫ u ⊥ ϕ t ( x , u ) d x d u = 2 c ( n + q − 1 ) n ω n I q ( K ) .

On the other hand, by (A4), when x is an interior point of K ∣ u ⊥ , we have

lim t → 0 ϕ t ( x , u ) = 2 q c X K ( x , u ) q − 1 h K ( ν K ( y ) ) u ⋅ ν K ( y ) − h K ( ν K ( y − ) ) u ⋅ ν K ( y − ) ,

where y and y − are the two boundary points of ∂ K ∩ ( x + R u ) . Since, for almost all u ∈ S n − 1 , ∂ K ∩ ( x + R u ) consists of at most two points, we obtain

∫ S n − 1 ∫ u ⊥ lim t → 0 ϕ t ( x , u ) d x d u = 2 q c ∫ S n − 1 ∫ K ∣ u ⊥ X K ( x , u ) q − 1 h K ( ν K ( y ) ) u ⋅ ν K ( y ) − h K ( ν K ( y − ) ) u ⋅ ν K ( y − ) d x d u = 2 q c ∫ S n − 1 ∫ ∂ K X K ( y , u ) q − 1 h K ( ν K ( y ) ) d ℋ n − 1 ( y ) d u .

By Lemmas A.4 and A.3, we obtain

I q ( K ) = q ( n + q − 1 ) n ω n ∫ S n − 1 ∫ ∂ K X K ( y , u ) q − 1 h K ( ν K ( y ) ) d ℋ n − 1 ( y ) d u .

Therefore, we obtain

∫ S n − 1 ∫ u ⊥ lim t → 0 ϕ t ( x , u ) d x d u = 2 c ( n + q − 1 ) n ω n I q ( K ) .

Thus, both sides of equation (A6) are equal to 2 c ( n + q − 1 ) n ω n I q ( K ) .□

We are now ready to prove Theorem 3.1.

Proof of Theorem 3.1

By Lemma A.5, there is a family of nonnegative integrable functions ϕ t ( x , u ) satisfying (A5) and (A6). Then, by the generalized dominated convergence theorem and Lemmas A.5, A.2, and A.3, we have

d d t t = 0 I q ( K t ) = 1 n ω n lim t → 0 ∫ S n − 1 ∫ u ⊥ 1 t ( X K t ( x , u ) q − X K ( x , u ) q ) d x d u = 1 n ω n ∫ S n − 1 ∫ u ⊥ lim t → 0 1 t ( X K t ( x , u ) q − X K ( x , u ) q ) d x d u = q n ω n ∫ S n − 1 ∫ K ∣ u ⊥ X K ( x , u ) q − 1 g ( ν K ( y ) ) u ⋅ ν K ( y ) − g ( ν K ( y − ) ) u ⋅ ν K ( y − ) d x d u = q n ω n ∫ S n − 1 ∫ ∂ K X K ( y , u ) q − 1 g ( ν K ( y ) ) d ℋ n − 1 ( y ) d u = 2 q ω n ∫ ∂ K V ˜ q − 1 ( K , z ) g ( ν K ( z ) ) d ℋ n − 1 ( z ) = ∫ S n − 1 g ( v ) d F q ( K , v ) = ∫ Ω g ( v ) d F q ( K , v ) ,

where y and y − are the two boundary points of ∂ K ∩ ( x + R u ) . Here, in the last equality, we used the fact that F q ( [ h 0 ] , ⋅ ) is concentrated on Ω .□

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Received: 2022-09-29

Accepted: 2022-11-27

Published Online: 2023-01-13

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

Keywords for this article

chord integral; chord measure; L p surface area measure; L p chord measure; L p Minkowski problem; L p chord Minkowski problem

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