Some logarithmic Minkowski inequalities for nonsymmetric convex bodies and related problems

where gK : ∂K → Sn–1 is the Gauss map of K , defined on ∂K , the set of boundary points of K that have a unique outer unit normal, and Hn–1 is an (n – 1)-dimensional Hausdorff measure, see, e.g., [5, 20, 24, 28]. The cone-volume measure of a convex body has clear geometric significance. Böröczky et al. in [4] posed the subspace concentration condition and completely solved the even Minkowski problem. The problem asks: What are the necessary and sufficient conditions on a finite Borel measure μ on Sn–1 such that μ is the conevolume measure of a convex body in Rn? In [30], Zhu solved the case of discrete measures whose supports are in general position. Uniqueness for the logarithmic Minkowski problem was completely settled for even measures in R2 in [3]. Recently, Stancu [23] proved the logarithmic Minkowski inequality for nonsymmetric convex bodies. Wang, Xu, and Zhou [25] gave the Lp version of Stancu’s results. For more results, see, e.g., [2, 4, 21, 25, 26, 30]. In his celebrated paper [17], Jerison solved the Minkowski problem for the capacitary measure, the measure that is the variational functional arising from the electrostatic capacity. Colesanti et al. in [9] extended Jerison’s work on electrostatic capacity to p-capacity.


Introduction
A convex body in an n-dimensional Euclidean space R n is a compact convex set that has nonempty interior. The cone-volume measure V K of a convex body K is a Borel measure on the unit sphere S n-1 defined for a Borel ω ∈ S n-1 by V K (ω) = 1 n x∈g -1 K (ω) x · g K dH n-1 (x), where g K : ∂K → S n-1 is the Gauss map of K , defined on ∂K , the set of boundary points of K that have a unique outer unit normal, and H n-1 is an (n -1)-dimensional Hausdorff measure, see, e.g., [5,20,24,28]. The cone-volume measure of a convex body has clear geometric significance. Böröczky et al. in [4] posed the subspace concentration condition and completely solved the even Minkowski problem. The problem asks: What are the necessary and sufficient conditions on a finite Borel measure μ on S n-1 such that μ is the conevolume measure of a convex body in R n ? In [30], Zhu solved the case of discrete measures whose supports are in general position. Uniqueness for the logarithmic Minkowski problem was completely settled for even measures in R 2 in [3]. Recently, Stancu [23] proved the logarithmic Minkowski inequality for nonsymmetric convex bodies. Wang, Xu, and Zhou [25] gave the L p version of Stancu's results. For more results, see, e.g., [2,4,21,25,26,30].
In his celebrated paper [17], Jerison solved the Minkowski problem for the capacitary measure, the measure that is the variational functional arising from the electrostatic capacity. Colesanti et al. in [9] extended Jerison's work on electrostatic capacity to p-capacity.
Naturally, the Minkowski problem for p-capacity was posed [9]: Given a finite Borel measure μ on S n-1 , what are the necessary and sufficient conditions on μ so that μ is the pcapacitary measure μ p (K, ·) of convex body K in R n ? The authors in [9] proved the uniqueness of the solution when 1 < p < n and existence and regularity when 1 < p < 2, the existence for 2 < p < n was solved by Akman et al. [1]. Inspired by the L p Minkowski problem for volume, Zou and Xiong [31] initiated the research into the L q Minkowski problem for p-capacitary measure. For more results, see, e.g., [6-8, 15-17, 19, 27, 29].

Main results
In this paper, we study the even logarithmic Minkowski problem and logarithmic Minkowski inequality for p-capacity. Our first result is to solve the existence part of the even logarithmic Minkowski problem for p-capacity. The problem asks: What are the necessary and sufficient conditions on a finite Borel measure μ on S n-1 such that μ is the L 0 pcapacitary measure of an origin-symmetric convex body in R n ? Our proof is based on the techniques in [4,9,23]. In order to solve the existence of the even logarithmic Minkowski problem, we use the definition of subspace concentration inequality in [4].

Definition 1.1 ([4])
A finite Borel measure μ on S n-1 is said to satisfy the subspace concentration inequality if, for every subspace ξ of R n such that 0 < dim ξ < n, The measure is said to satisfy the subspace concentration condition if, in addition to satisfying the subspace concentration inequality (1.1), whenever for some subspace ξ , then there exists a subspace ξ , which is complementary to ξ in R n , so that Equality holds if and only if K and L are homothetic.
h L (u) . We obtain the third result.
Equality holds if and only if K = L.
In general, we have This paper is organized as follows. In Sect. 2, we give a minimization problem for pcapacity. In Sect. 3, we give the proof of Theorem 1.1. In Sect. 4, we prove Theorem 1.2 and its application.

Preliminaries
For quick reference, we collect some basic facts on the theory of convex bodies. Good references are the books by Schneider [22].
Denote by K n the set of convex bodes in R n and by K n o the set of convex bodies with the origin o in its interiors. Let h K and h L be the support functions (h K (u) = h(K, u) := max u∈S n-1 {x · u : x ∈ K}, where x · u denotes the inner product of u and x) of K . Let I ⊂ R be an interval containing 0, C + (S n-1 ) be a class of continuous and positive functions on S n-1 , C + e (S n-1 ) means the subsets of C + (S n-1 ) are the even functions, and assume that h t (u) = h(t, u) : I × S n-1 → (0, +∞) is continuous. Put The convex body Ω t is called the Aleksandrov body associated with h t . Via the support function, Böröczky et al. in [3] defined the log Minkowski combination (1λ) · K + 0 λ · L, that is, (1. 2) The surface area measure S K of a convex body K is a Borel measure on the unit sphere S n-1 defined for a Borel ω ∈ S n-1 by Let Ω be a bounded convex domain in R n , n ≥ 3, Ω be its closure. The equilibrium potential u = u Ω is the unique solution to the boundary value problem where p is the p-Laplace operator and 1 < p < n. Due to Dahlbeg [12], we can see that ∇u has non-tangential limits almost everywhere on ∂Ω and |∇u| ∈ L p (∂Ω, H n-1 ). The p-capacity was defined by In [9], the authors proved the following Hadamard variational formula for p-capacity. Let K, L ∈ K n o and 1 < p < n, |∇u K | p dH n-1 , and the Poincaré p-capacity formula

A minimization problem
In this section, we study a minimization problem, its solution also solves the logarithmic Minkowski problem for p-capacity. The following lemma will be needed.
With Lemma 2.1 in hand, we use the definition of log Minkowski combination to prove the following result.
Then, by Lemma 2.1, we obtain .
In view of the above result, we give the following definitions.
for ω ⊂ S n-1 , is called the L 0 p-capacitary measure of K . Definition 2.2 Let 1 < p < n and K, L ∈ K n o , then L 0 mixed p-capacity of K and L is defined by The next lemma shows that C p,0 (K, L) is continuous in (K, L).
uniformly on S n-1 . By Definition 2.2, the desired limit is obtained.
The weak convergence of p-capacitary measure implies the weak convergence of μ p,0 as follows.
We now consider the minimization problem where K n e is an origin-symmetric convex body with nonempty interior, μ is a finite even Borel measure on S n-1 with total mass |μ| > 0, and the logarithmic functional Φ μ : K n e → R is defined by The following theorem shows a solution to the logarithmic Minkowski type problem for the measure μ is a solution to minimization problem for the function Φ μ . Theorem 2.1 Let μ be a finite even Borel measure on S n-1 with |μ| > 0 and 1 < p < n. If K is an origin-symmetric convex body such that C p (K) = |μ| and then the measure μ is the L 0 p-capacitary measure of K .
Proof Clearly, we may assume that μ is a probability measure. Next, we consider the minimization problem where the continuous functional F : C + e (S n-1 ) → (0, +∞) by here K f denotes the Wulff shape of f ∈ C + e (S n-1 ). Notice that F is homogeneous of degree 0, i.e., F(sf ) = F(f ) for s > 0. By the properties of Aleksandrov body, we have h K f ≤ f . According to Lemma 2.1 in [31], we have . Therefore, we shall search for the infimum of F among the support functions of origin-symmetric convex bodies. It follows that the infimum of F Obviously, the right infimum is attained at K ∈ K n e . Thus, the support function h K > 0 is a solution of minimization problem, i.e., Given the function h t = h(·, t) : S n-1 × R → (0, ∞) is defined by by the function F(h t ) has a minimum at t = 0, this implies that On the other hand, it follows from C p (K) = 1 that Since f ∈ C + e (S n-1 ) is arbitrary, we conclude that this completes the proof of the theorem.

Logarithmic Minkowski problem
In the previous section, we have proved the existence of a solution to the logarithmic Minkowski problem by using the variational argument. In this section, we show the proof of the main result. Proof Without loss of generality, we assume that |μ| = 1. Let Q l be the minimizing sequence of origin-symmetric convex bodies, that is, Q l satisfies the C p (Q l ) = 1 and Taking L = γ -1 n-p B n , γ = ( p-1 n-p ) 1-p ω n , here B n is a unit ball. Then C p (L) = 1, it follows that By John's theorem [18] associated with Q l , there exists an origin-symmetric ellipsoid E l so that Note that, for every origin-symmetric ellipsoid E l , there is a cross-polytope P l denoted by P l = [±a 1l u 1l , . . . , ±a nl u nl ] such that P l ⊆ E l ⊆ √ nP l . Hence, P l ⊆ Q l ⊆ nP l . It follows from C p (Q l ) = 1 that C p (P l ) ≥ n -1 n-p . We next claim that suppose K is a convex body containing the origin and satisfying V (K) = 0, then C p (K) = 0. In fact, since It follows that S(K, {h K > 0}) = 0, which implies H n-1 (g -1 K ({h K > 0})) = 0. Together with the fact that |∇u| p is integrable on ∂K , we have By (1.5), we have C p (K) = 0. Thus, there exists a constant c > 0 such that V (K) ≥ cC p (K). This yields n i=1 a il = n!V (P l ) 2 n ≥ cn!C p (P l ) 2 n := γ 1 .
Assume that Q l is not bounded. Then P l is not bounded, thus there exists a nl such that lim l→∞ a nl = ∞. Applying Lemma 6.2 in [3] to P l = γ -1 n 1 P l implies that Φ μ (P l ) is not bounded, which gives that Φ μ (Q l ) is not bounded. This contradicts (3.1). Therefore, the sequence Q l is bounded. By Blaschke's selection theorem, Q l has a subsequence that converges to an origin-symmetric convex body K . In the following, we prove K ∈ K n o is ndimensional. Let dim K ≤ n -2. Since 1 < p < 2, we have dim K ≤ n -2 < np, so C p (K) = 0 by [13], p. 179, which contradicts that C p (K) = 1. Let dim K = n -1, there exists a unit vector u ∈ S n-1 such that K ⊂ u ⊥ . Then u, -u ∈ supp μ. But μ satisfies μ({-u}) = 0 whenever μ({u}) > 0 for any u ∈ S n-1 , which is a contradiction. This gives the desired result.
We get the proof of Theorem 1.1 directly from Theorem 2.1 and Lemma 3.1.

The reverse Hölder inequality gives
Equality holds if and only if K is homothetic to L. This gives the first inequality of Theorem 1.2. Now, we prove the second inequality of Theorem 1.2. Using Minkowski's inequality for p-capacity [10], we have Equality holds if and only if K is homothetic to L.
Similarly, we obtain the reverse form of Theorem 1.2.
Theorem 4.1 Let K, L ∈ K n o and 1 < p < n. Then with equality if and only if K is homothetic to L.
First proof Via the same idea in Theorem 1.2, we have From the reverse Hölder inequality, we have Thus, Second proof We will use Gibbs' inequality [11], i.e., let f and g be the probability density functions on a measure space (X, ν), then f ln f dν ≥ f ln g dν, equality holds if and only if f = g. On the one hand, let f dν(·) = h L h K C p (L) dC p (L, K, ·) and g dν(·) = 1 C p (L,K) dC p (L, K, ·). By Gibbs' inequality in [11], we have which implies that According to the equality condition, we know that K is homothetic to L.
On the other hand, we set f dν(·) = 1 C p (L,K) dC p (L, K, ·) and g dν(·) = h L h K C p (L) dC p (L, K, ·), then which is equivalent to the following inequality: with equality if and only if K is homothetic to L. This completes the proof of the theorem.
Moreover, we show the logarithmic Minkowski type inequality for p-capacity.
Proof of Theorem 1.3 Let q ∈ R and 1 < p < n, consider the function We now claim that G(q) is a log-convex function. In fact, let t ∈ (0, 1), by Hölder's inequality, we get that Applying the Hadamard type inequality in [14], we have From Fubini-Tonelli's theorem, we obtain .
Notice that Thus, by Theorem 1.2, we have Suppose G(q) ≡ 0, then h K (u) = h L (u) for almost all u in L 0 p-capacitary measure of L, or equivalently with respect to the p-capacitary measure of L. This implies that C p (L, K) = C p (L). According to the equality condition of Minkowski inequality for p-capacity and L ⊆ K , we obtain K = L. Assume that K, L ∈ K n o are arbitrary and L is not included in K , then there exists 0 < t < 1 such that tL ⊆ K . By (4.4), we have which is equivalent to the following inequality: Taking t = min u∈supp C p (K,·) h L (u) , we obtain the second inequality, with equality if and only if K and L are homothetic.
Obviously, Theorem 1.3 implies the following corollary. Equality holds if and only if K = tL.
Finally, we shall give an application of Theorem 1.2. Let F n be the set of all convex bodies with positive continuous curvature functions and L be a convex body in F n with a curvature function f L . To simplify the notation, we write K * = {x ∈ R n : x · y ≤ 1, y ∈ K} to be the polar body of K and Φ p (L) = S n-1 f n n+1 L |∇u L | np n+1 dS(u). Theorem 4.2 If K ∈ K n o , 1 < p < n, and L ∈ K n o ∩ F n , then where c(p, K, L) := (exp( S n-1 ln( h K (u) h L (u) )) dC * p (L, K, u)) n .
Proof Let K and L be distinct, by Theorem 1.2, we have Using the reverse Hölder inequality, we obtain This gives the desired inequality. For the case K = L ∈ K n o ∩ F n , according to the above proof, we also obtain the desired inequality.
Taking K = L, we obtain the following result. Corollary 4.2 If K ∈ K n o ∩ F n and 1 < p < n, then