# The Functional Orlicz Brunn-Minkowski Inequality forq-Capacity

Authors: Wei Wang; Juan Li; Rigao He; Lijuan Liu
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we establish functional forms of the Orlicz Brunn-Minkowski inequality and the Orlicz-Minkowski inequality for the electrostaticq-capacity, which generalize previous results by Zou and Xiong. We also show that these two inequalities are equivalent.


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## Body

## 1. Introduction

The classical Brunn-Minkowski inequality was inspired by questions around the isoperimetric problem. It is viewed as one of cornerstones of the Brunn-Minkowski theory, which is a beautiful and powerful tool to conquer all sorts of geometrical problems involving metric quantities such as volume, surface area, and mean width.An excellent reference on this inequality is provided by Gardner [1].In 2015, Colesanti, Nyström, Salani, Xiao, Yang, and Zhang (CNSXYZ) [2] introduced the electrostatic q-capacity. Let K be a compact set in the n-dimensional Euclidean space Rn. For 1<q<n, the electrostatic q-capacity, CqK, of K is defined by
(1)CqK=inf∫Rn∇uqdx:u∈Cc∞Rnandu≥χK,where Cc∞Rn denotes the set of functions from C∞Rn with compact supports and χK is the characteristic function of K. If q=2, then C2K is the classical electrostatic (or Newtonian) capacity of K. The Minkowski-type problems for the electrostatic q-capacity have attracted increasing attention [2–10]. The electrostatic q-capacity also has applications in analysis, mathematical physics, and partial differential equations (see [11–13]).The electrostaticq-capacity can be extended on function spaces. Let CSn−1 denote the set of continuous functions defined on Sn−1, which is equipped with the metric induced by the maximal norm. Write C+Sn−1 for the set of strictly positive functions in CSn−1. For 1<q<n and f∈C+Sn−1, define the electrostatic q-capacity Cqf by
(2)Cqf=Cqf,where f denotes the Aleksandrov body (also known as the Wulff shape) associated with f. For nonnegative f∈CSn−1, the Aleksandrov body f is defined by
(3)f=∩ξ∈Sn−1x∈Rn:x⋅ξ≤fξ.Obviously,f is a compact convex set containing the origin and hf≤f, where hf denotes the support function of f. Moreover,
(4)K=hK,for every compact convex set K containing the origin. If f∈C+Sn−1, thenf is a convex body in Rn containing the origin in its interior. The Aleksandrov convergence lemma reads: if the sequence fjj⊂C+Sn−1 converges uniformly to f∈C+Sn−1, then, limj→∞fj=f.Supposea,b≥0 (not both zero). If f,g∈C+Sn−1, then, the Lp Minkowski sum a⋅g+pb⋅f is defined by
(5)a⋅f+pb⋅g=afp+bgp1/p,where the Lp scalar multiplication a⋅f is defined by aa/pf. By the definition of the Aleksandrov body (3), we have a⋅hK+pb⋅hL=a⋅K+pb⋅L for convex bodies K and L containing the origin in their interiors. Here, a⋅K+pb⋅L denotes the Lp Minkowski sum of K and L, i.e.,
(6)ha⋅K+pb⋅Lp=ahKp+bhLp,for every u∈Sn−1, which was defined by Firey [14]. In the mid 1990s, it was shown in [15, 16] that when Lp Minkowski sum is combined with volume the result is an embryonic Lp-Brunn-Minkowski theory. Zou and Xiong ([7], Theorem 3.11) established the functional form of the Lp Brunn-Minkowski inequality for the electrostatic q-capacity. Suppose 1<p<∞ and 1<q<n.Iff,g∈C+Sn−1, then
(7)Cqf+pgp/n−q≥Cqfp/n−q+Cqgp/n−q,with equality if and only if f and g are dilates.The Orlicz Brunn-Minkowski theory which was launched by Lutwak et al. in a series of papers [17–19] is an extension of the Lp Brunn-Minkowski theory. This theory has been considerably developed in the recent years. The Orlicz sum was introduced by Gardner et al. [20]. Let Φ be the class of convex, strictly increasing functions, ϕ:0,∞⟶0,∞ with ϕ0=0. Suppose ϕ∈Φ and a,b≥0 (not both zero). If K and L are convex bodies that contain the origin in their interiors in Rn, then, the Orlicz sum a⋅K+ϕb⋅L is the convex body defined by
(8)ha⋅K+ϕb⋅Lu=infτ>0:aϕhKuτ+bϕhLuτ≤ϕ1,for every u∈Sn−1. Gardner et al. ([20], Corollary 7.5) established the Orlicz Brunn-Minkowski inequality (see also ([21], Theorem 1). Same as the Orlicz sum of convex bodies, we extend the Lp Minkowski sum of functions to the Orlicz sum. For ϕ∈Φ, f,g∈C+Sn−1, and a,b≥0 (not both zero), the Orlicz sum a⋅f+ϕb⋅g is defined by
(9)a⋅f+ϕb⋅g=infτ>0:aϕfτ+bϕgτ≤ϕ1.If we takeϕt=tpp≥1 in (9), then it, induces the Lp Minkowski sum (5). By the definition of the Aleksandrov body (3), (8), (9), and (4), we have a⋅hK+ϕb⋅hL=a⋅K+ϕb⋅L for convex bodies K and L containing the origin in their interiors.The main aim of this paper is to establish the functional form of the Orlicz Brunn-Minkowski inequality for the electrostaticq-capacity.Theorem 1.
Supposeϕ∈Φ and 1<q<n. If f,g∈C+Sn−1, then,
(10)ϕCqfCqf+ϕg1/n−q+ϕCqgCqf+ϕg1/n−q≤ϕ1.
Ifϕ is strictly convex, equality holds if and only if f and g are dilates.


## 2. Notation and Preliminary Results

For excellent references on convex bodies, we recommend the books by Gardner [22], Gruber [23], and Schneider [24].We will work inRn equipped with the standard Euclidean norm. Let x⋅y denote the standard inner product of x,y∈Rn. For x∈Rn, x=x⋅x denotes the Euclidean norm of x. We write B=x∈Rn:x≤1 and Sn−1 for the standard unit ball of Rn and its surface, respectively. Each compact convex set K is uniquely determined by its support function hK:Rn⟶R, which is defined by hKx=maxx⋅y:y∈Rn, for x∈Rn. Obviously, the support function is positively homogeneous of order 1.The class of compact convex sets inRn is often equipped with the Hausdorff metric δH, which is defined for compact convex sets K and L by
(11)δHK,L=maxhKu−hLu:u∈Sn−1.Denote byKn the set of convex bodies in Rn and by Kon the set of convex bodies which contain the origin in their interiors. For s>0, the set sK=sx:x∈K is called a dilate of convex body K. Convex bodies K and L are said to be homothetic, provided K=sL+x for some s>0 and x∈Rn. Let K,L∈Kn, the Minkowski sum of K and L is the convex body
(12)K+L=x,y:x∈K,y∈L.Some properties of the electrostaticq-capacitary measure are required [2, 3, 7, 8, 11]. The electrostatic q-capacitary measure, μqE,⋅, of a bounded open convex set E in Rnis the measure on the unit sphere Sn−1 defined for ω⊂Sn−1 and 1<q<n by
(13)μqE,ω=∫g−1ω∇UqdHn−1,where g−1:Sn−1⟶∂E (the set of boundary points of E) denotes the inverse Gauss map, Hn−1 the n−1-dimensional Hausdorff measure, and U the q-equilibrium potential of E. If K∈Kn, then the electrostatic q-capacitary measure μqK,⋅ has the following properties. First, it is positively homogeneous of degree n−q−1, i.e., μqsK,⋅=sn−q−1μqK,⋅ for s>0. Second, it is translation invariant, i.e., μqK+x,⋅=μqK,⋅ for x∈Rn. Third, its centroid is at the origin, i.e., ∫Sn−1udμqK,u=0. Moreover, it is absolutely continuous with respect to the surface area measure SK,⋅. The weak convergence of the electrostatic q-capacitary measure is proved by CNSXYZ ([2], Lemma 4.1): if Kj⊂Kn converges to K∈Kn, thenμqKj,⋅ converges weakly to μqK,⋅.CNSXYZ [2] showed the Hadamard variational formula for the electrostatic q-capacity: for K,L∈Kn and 1<q<n,
(14)dCqK+tLdtt=0+=q−1∫Sn−1hLudμqK,u.And variational formula (14) leads to the following Poincare q-capacity formula:
(15)CqK=q−1n−q∫Sn−1hKudμqK,u.The electrostaticq-capacity CqK has the following properties. First, it is increasing with respect to set inclusion; that is, if K1⊆K2, then CqK1≤CqK2. Second, it is positively homogeneous of degree n−q, i.e., CqsK=sn−qCqK for s>0. Third, it is a rigid motion invariant, i.e.,CqφK+x=CqK for x∈Rn and φ∈On. If q=2, then (15) induces the Poincare capacity formula
(16)C2K=1n−2∫Sn−1hKudμ2K,u.LetCSn−1 denote the set of continuous functions defined on Sn−1, which is equipped with the metric induced by the maximal norm. Write C+Sn−1 for the set of strictly positive functions in CSn−1. Let K∈Kon and g∈CSn−1. There is a t0>0 such that hK+tg∈C+Sn−1 for t<t0. The Aleksandrov body hK+tg is continuous in t∈−t0,t0. The Hadamard variational formula for the electrostatic q-capacity [2] states the following:
(17)dCqhK+tgdtt=0+=q−1∫Sn−1gudμqK,u.Forf∈C+Sn−1, define
(18)Cqf=Cqf.Obviously,CqhK=CqK for every K∈Kon. By the Aleksandrov convergence lemma and the continuity of Cq on Kn, the functional Cq:C+Sn−1⟶0,∞ is continuous. For K∈Kon and g∈CSn−1, the mixed electrostatic q-capacity CqK,g is defined by
(19)CqK,g=1n−qdCqhK+tgdtt=0+.Applying the Hadamard variational formula for the electrostaticq-capacity, the mixed electrostatic q-capacity CqK,g has the following integral representation:
(20)CqK,g=q−1n−q∫Sn−1gudμqK,u.LetL∈Kn. If g=hL, then, CqK,g is the mixed electrostatic q-capacity CqK,L, which has the following integral representation:
(21)CqK,L=q−1n−q∫Sn−1hLudμqK,u.The Minkowski inequality for the electrostaticq-capacity ([2], Theorem 3.6) states the following: let1<q<n.IfK,L∈Kn, then,
(22)CqK,L≥CqKn−q−1/n−qCqL1/n−q,with equality if and only if K and L are homothetic.Let1≤p<∞ and 1<q<n. For K∈Kon and g∈CSn−1, the Lp Hadamard variational formula for the electrostatic q-capacity [7] states the following:
(23)dCqhK+pt⋅gdtt=0+=q−1p∫Sn−1guphKu1−pdμqK,u.TheLp mixed electrostatic q-capacity Cp,qK,g is defined by
(24)Cp,qK,g=1n−qdCqhK+pt⋅gdtt=0+=q−1n−q∫Sn−1guphKu1−pdμqK,u.Takeg=hK in (24), and combine CqK,g=CqK to obtain the Poincare q-capacity formula (15). Zou and Xiong ([7], Theorem 3.9) established the Lp Minkowski inequality for the Lp electrostatic q-capacity: let 1<p<∞ and 1<q<n. If K∈Kon and g∈CSn−1, then,
(25)CqK,g≥CqKn−q−p/n−qCqLp/n−q,with equality if and only if K and g are dilates.Based on the Orlicz sum (9), we define the Orlicz mixed electrostatic q-capacity as follows. For K∈Kon and g∈CSn−1, the Orlicz mixed electrostatic q-capacity Cϕ,qK,g is defined by
(26)Cϕ,qK,g=1n−qdCqhK+ϕt⋅gdtt=0+.Indeed, the Orlicz mixed electrostaticq-capacity can be extended on function spaces. Let ϕ∈Φ and 1<q<n. For f∈C+Sn−1 and g∈CSn−1, the Orlicz mixed electrostatic q-capacity Cϕ,qf,g is defined by
(27)Cϕ,qf,g=1n−qdCqf+ϕt⋅gdtt=0+.Iff=hK with K∈Kon, then, Cϕ,qhK,g=Cϕ,qK,g. If g=hL with L∈Kon, then, Cϕ,qf,hL is the Orlicz mixed electrostatic q-capacity Cϕ,qf,L. In particular, Cϕ,qhK,hK=ϕ1CqK for every K∈Kon.


## 3. Main Results

The following variational formula of electrostaticq-capacity plays a crucial role in our proof.Lemma 2 ([2], Lemma 5.1).
LetI⊂R be an interval containing 0 in its interior, and let htu=ht,u:I×Sn−1⟶0,∞ be continuous such that the convergence in
(28)h′0,u=limt→0ht,u−h0,utis uniformly on Sn−1. Then,
(29)dCqhtdtt=0+=q−1∫Sn−1h′0,udμqh0,u.
Supposeϕ∈Φ, f,g∈CqSn−1, and a,b≥0 (not both zero). For every given u∈Sn−1, the function t↦aϕfu/t+bϕgu/t is strictly decreasing. By the definition of the Orlicz sum (9), we have a⋅f+ϕb⋅gu=t and only if aϕfu/t+bϕgu/t=ϕ1.
for everyu∈Sn−1.The continuity properties of the Orlicz sum were established by Xi et al. [21].Lemma 3 ([21], Lemma 3.1).
Suppose,ϕ∈Φf∈C+Sn−1, g∈CSn−1, and a,b≥0 (not both zero).
(i)
Letfi,gi⊂C+Sn−1 and gi⊂CSn−1 such that fi⟶f and gi⟶g, respectively. Then, a⋅fi+ϕb⋅gi⟶a⋅f+ϕb⋅g(ii)
Letϕi⊂Φ such that ϕi→ϕ. Then, a⋅f+ϕib⋅g⟶a⋅f+ϕb⋅g(iii)
Letai,bi≥0 (not both zero) such that ai⟶a and bi⟶b. Then(30)ai⋅f+ϕbi⋅g⟶a⋅f+ϕb⋅gDue to Lemma2, the integral representation of the Orlicz mixed electrostatic q-capacity is given.Lemma 4.
Supposeϕ∈Φ and 1<q<n. If f∈C+Sn−1 and g∈CSn−1, then
(31)Cφ,qf,g=q−1n−q∫Sn−1φgufufudμqf,u.Proof.
Take an intervalI=0,t0 for 0<t0<∞. Denote htu:I×Sn−1⟶0,∞ by
(32)htu=ht,u=f+ϕt⋅gu.
Then, the definition of the Orlicz sum (9) and Lemma 3 imply that the function htu:I×Sn−1⟶0,∞ is continuous. By (9), we have
(33)ϕfuhtu+tϕguhtu=ϕ1,for every u∈Sn−1. Since φ∈ϕ, we obtain
(34)dhtdt=ht2ϕg/htfϕ′f/ht+tfϕ′f/ht.
Note thatf/h⟶1− as t⟶0+ and the fact that h0=f. Thus,
(35)limt→0+ht−h0t=fϕg/fϕl′1,uniformly on Sn−1, where ϕl′1 denotes the left derivative of ϕt at t=1. Apply Lemma 2 and (35) to get
(36)dCqf+ϕt⋅gdtt=0+=q−1φl′1∫Sn−1ϕgufudμqf,u.
Thus, (27) and (36) yield the desired lemma.Indeed, (36) can be considered as the Orlicz Hadamard variational formula for the electrostatic q-capacity. If we take ϕt=tp1≤p<∞ and f=hK with K∈Kon in (36), then, we obtain the Lp Hadamard variational formula (23).Note thathK=K for every K∈Kon. Take f=hK in Lemma 4 to getLemma 5.
Supposeϕ∈Φ and 1<q<n. If K∈Kon and g∈CSn−1, then,
(37)Cϕ,qK,g=q−1n−q∫Sn−1ϕguhKuhKudμqK,u.
A direct consequence of Lemma4 and the homogeneity of the electrostatic q-capacitary measure can be obtained.Corollary 6.
Supposeϕ∈Φ and 1<q<n. If f∈C+Sn−1, then
(38)Cφ,qcf,f=φ1ccn−qCqf=φ1ccn−qCqf,
for everyc>0.Letϕ∈Φ, 1<q<n, and K,L∈Kon. Note that K+ϕt⋅L=hK+ϕt⋅hL, and apply (18) and (36) to obtain
(39)dCqK+ϕt⋅Ldtt=0+=q−1ϕl′1∫Sn−1ϕhLuhKuhKudμqK,u.Based on (39), one can define the Orlicz mixed electrostatic q-capacity Cϕ,qK,L of convex bodies K and L as follows:
(40)Cϕ,qK,L=q−1n−q∫Sn−1ϕhLuhKuhKudμqK,u,which was first defined by Hong et al. ([10], Definition 3.1).Lemma 7.
Supposeϕ∈Φ, f∈C+Sn−1, g∈CSn−1, and 1<q<n.
(i)
Letϕ1,ϕ2∈Φ. If ϕ1≤ϕ2, then Cϕi,qf,g⟶Cϕ2,qf,g(ii)
Letfi⊂C+Sn−1 and gi⊂CSn−1 such that fi⟶f and gi⟶g, respectively. Then, Cϕ,qfi,gi⟶Cϕ,qf,g(iii)
Letϕi⊂Φ such that ϕi⟶ϕ. Then, Cϕi,qf,g⟶Cϕ,qf,gProof.
It follows from (31) that (i) holds if ϕ1≤ϕ2.
Sincef>0,fi>0,g≥0,gi≥0 and fi⟶f, gi⟶g uniformly on Sn−1; it follows that gi/fi⟶g/f uniformly on Sn−1. Note that ϕ∈Φ, we have ϕgi/fifi⟶ϕg/ff uniformly on Sn−1. The Aleksandrov convergence lemma implies that fi⟶f uniformly on Sn−1. Meanwhile, the convergencefi⟶f implies that μqfi,⋅⟶μqf,⋅ weakly. Applying Lemma 4, one concludes that (ii) holds.
Clearly, there exists a compact intervalI⊂0,∞ such that g/f∈I for all u∈Sn−1.
(iii) directly follows from Lemma4 and the fact that the sequence ϕit converges uniformly to ϕt on I.
Next, we show that there is a natural Orlicz extension of the Minkowski inequality for the electrostaticq-capacity.Theorem 8.
Supposeϕ∈Φ and 1<q<n. If f,g∈C+Sn−1, then,
(41)Cϕ,qf,gCqf≥ϕCqgCqf1/n−q.
Ifφ is strictly convex, then equality holds if and only if f and g are dilates.Proof.
By the definition of the mixed electrostaticq-capacity (20) and the fact that hg≤g, we have
(42)Cqf,g=Cqf,hg≤Cqf,g.for every f,g∈C+Sn−1. From (31), Jensen’s inequality, (20), (42), (22), and (18), it follows that
(43)Cϕ,qf,gCqf=q−1/n−q∫Sn−1ϕgu/fufudμqf,uCqf≥ϕCqf,gCqf≥ϕCqf,gCqf≥ϕCqg1/n−qCqf1/n−q=ϕCqgCqf1/n−q.
It remains to prove the equality condition. Now, supposeϕ is strictly convex. If equality in (41) holds, then, by the equality condition of Jensen’s inequality, there exists an s>0 such that g=sf for almost every u∈Sn−1with respect to the measure f⋅dμqf,⋅/Cqf. Then, we have
(44)s=q−1/n−q∫Sn−1gu/fufudμqf,uCqf=Cqf,gCqf=Cqf,gCqf,where the last step is from the equality condition of (42). The definition of Aleksandrov body implies that hg=shf for almost every u∈Sn−1 with respect to the measure f⋅dμqf,⋅/Cqf. Thus,
(45)Cqf,ghfu=Cqfhgu,for almost every u∈Sn−1 with respect to the measure f⋅dμqf,⋅/Cqf. By the equality condition of the Minkowski inequality for the electrostatic q-capacity, there exists x∈Rn such that g=sf+x.
Hence, for almost everyu∈Sn−1 with respect to the measure f⋅dμqf,⋅,
(46)sCqf+q−1n−qx⋅∫sn−1⋅udμqf,uhfu=Cqfshfu+x⋅u.
Since the centroid ofμqf,⋅ is at the origin, we have that x⋅u=0 for almost every u∈Sn−1 with respect to the measure f⋅dμqf,⋅/Cqf. Note that the electrostatic q-capacitary measure μqf,⋅ is not concentrated on any great subsphere of Sn−1. Hence, x=0, which in turn implies that f and g are dilates.
Conversely, assume thatf and g are dilates, say, f=cg for some c>0. From our assumption, Corollary 6, (18), and the fact that Cqcg=cn−qCqg, it follows that
(47)Cϕ,qf,gCqf=Cϕ,qcg,gCqcg=ϕ1/ccn−qCqgcn−qCqg=ϕ1c=ϕCqgCqf1/n−q.
This completes the proof.By using the Orlicz-Minkowski inequality for the electrostaticq-capacity, we establish the following Orlicz Brunn-Minkowski inequality for the electrostatic q-capacity which is the general version of Theorem 1.Theorem 9.
Supposeϕ∈Φ and 1<q<n. If f,g∈C+Sn−1 and a,b≥0 (not both zero); then,
(48)aϕCqfCqa⋅f+ϕb⋅g1/n−q+bϕCqfCqa⋅f+ϕb⋅g1/n−q≤ϕ1.
Ifϕ is strictly convex, then equality holds if and only if f and g are dilates.Proof.
By (31), (9), and the Orlicz-Minkowski inequality for the electrostatic q-capacity (41), we have
(49)ϕ1=Cϕ,qa⋅f+ϕb⋅g,a⋅f+ϕb⋅gCqa⋅f+ϕb⋅g=aCϕ,qa⋅f+ϕb⋅g,fCqa⋅f+ϕb⋅g+bCϕ,qa⋅f+ϕb⋅g,gCqa⋅f+ϕb⋅g≥aϕCqfCqa⋅f+ϕb⋅g1/n−q+bϕCqgCqa⋅f+ϕb⋅g1/n−q.
By the equality condition of the Orlicz-Minkowski inequality for the electrostaticq-capacity, we have that if ϕ is strictly convex, then equality in (48) holds if and only if f and g are dilates of a⋅f+φb⋅g.Remark 1.
The caseϕt=tp1≤p<∞ of Theorem 9 was established by Zou and Xiong [7].ForK,L∈Kon, take f=hK and g=hL in Theorem 9 to obtain the following Orlicz-Brunn-Minkowski inequality for the electrostatic q-capacity, which was established by Hong et al. [10].Corollary 10 ([10], Theorem 4.2).
Supposeϕ∈Φ and 1<q<n. IfK,L∈Kon, then
(50)aϕCqKCqa⋅K+ϕb⋅L1/n−q+bϕCqLCqa⋅K+ϕb⋅L1/n−q≤ϕ1.
Ifφ is strictly convex, then equality holds if and only if K and L are dilates.Remark 2.
The caseφt=t of Corollary 10 was obtained by Colesanti and Salani [25]. Borell [26] first established the Brunn-Minkowski inequality for the classical electrostatic capacity, while its equality condition was shown by Caffarelli et al. [4].Theorem 11.
Supposeϕ∈Φ, 1<q<n, and f,g∈C+Sn−1. Then, the Orlicz-Brunn-Minkowski inequality for the electrostatic q-capacity implies the Orlicz-Minkowski inequality for the electrostatic q-capacity.Proof.
Fort≥0 and f,g∈C+Sn−1, define the function Gt by
(51)Gt=ϕ1−ϕCqfCqf+ϕt⋅g1/n−q−tϕCqgCqf+ϕt⋅g1/n−q.
The Orlicz-Brunn-Minkowski inequality for the electrostaticq-capacity implies that Gt is nonnegative. Obviously, G0=0. Thus,
(52)limt→0+Gt−G0t≥0.
On the other hand, by (51) and the continuity of Cq, we have
(53)limt→0+Gt−G0t=limt→0+ϕ1−ϕCqf/Cqf+ϕt⋅g1/n−q−tϕCqf/Cqf+ϕt⋅g1/n−qt=limt→0+ϕ1−ϕCqf/Cqf+ϕt⋅g1/n−qt−ϕCqgCqf1/n−q=limt→0+ϕ1−ϕCqf/Cqf+ϕt⋅g1/n−q1−Cqf/Cqf+ϕt⋅g1/n−q⋅1−Cqf/Cqf+ϕt⋅g1/n−qt−ϕCqgCqf1/n−q.
Lets=Cqf/Cqf+ϕt⋅g1/n−q. Note that s⟶1− as t⟶0+. Consequently,
(54)limt→0+ϕ1−ϕCqf/Cqf+ϕt⋅g1/n−q1−Cqf/Cqf+ϕt⋅g1/n−q=ϕl′1.
The continuity ofCq and (27) imply
(55)limt→0+1−Cqf/Cqf+ϕt⋅g1/n−qt=limt→0+Cqf+φt⋅g−Cqf1/n−qt⋅limt→0+Cqf+ϕt⋅g−1/n−q=1n−qCqf1/n−q−1⋅limt→0+Cqf+ϕt⋅g1/n−q−Cqft⋅Cqf−1/n−q=Cϕ,qf,gϕl′1Cqf.
From (53), (54), (55), and (52), it follows that
(56)limt→0+Gt−G0t=Cϕ,qf,gCqf−ϕCqgCqf1/n−q≥0,which implies the Orlicz-Minkowski inequality for the electrostatic q-capacity (41).
Finally, we show an immediate application of the Orlicz-Minkowski inequality for the electrostaticq-capacity.Lemma 12.
Supposeϕ∈Φ and 1<q<n. If f,g∈C+Sn−1 and C is a subset of C+Sn−1 such that f,g∈C, then the following assertions hold:
(i)
Cϕ,qh,f=Cϕ,qh,g for all h∈C; then f=g(ii)
Cϕ,qf,h/Cqf=Cϕ,qg,h/Cqf for all h∈C; then f=gProof.
We first show that (i) holds. SinceCϕ,qf,f=ϕ1Cqf, it follows that ϕ1=Cϕ,qf,g/Cqg by the assumption. By the Orlicz-Minkowski inequality for the electrostatic q-capacity, we have ϕ1≥ϕCqf/Cqg1/n−q. The monotonicity of ϕ and 1<q<n imply that
(57)CqfCqg<1,with equality if and only if f and g are dilates. This inequality is reversed if interchanging f and g. So,Cqf=Cqg and f and g are dilates. Assume that sf=sg for some s>0. The homogeneity of Cq implies s=1. Thus, f=g.
Then, we can prove (ii) with the similar arguments in (i).If the Orlicz mixed electrostaticq-capacity Cϕ,q is restricted on convex bodies, then we obtain the following characterizations for identity of convex bodies, which were proved by Hong et al. [13].Corollary 13 ([10], Theorem 3.3).
Supposeϕ∈Φ and 1<q<n. If K,L∈Kon and C is a subset of Kon such that K,L∈C, then the following assertions hold:
(i)
Cϕ,qQ,K=Cϕ,qQ,L for all Q∈C; then K=L(ii)
Cϕ,qK,Q/CqK=Cϕ,qL,Q/CqLfor all Q∈C; then K=L



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*Source: 1670617-2020-10-08.xml*