The Functional Orlicz Brunn-Minkowski Inequality for q-Capacity

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, CqðKÞ, of K is defined by


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 R n . For 1 < q < n, the electrostatic q-capacity, C q ðKÞ, of K is defined by where C ∞ c ðR n Þ denotes the set of functions from C ∞ ðR n Þ with compact supports and χ K is the characteristic function of K. If q = 2, then C 2 ðKÞ is the classical electrostatic (or Newtonian) capacity of K. The Minkowski-type problems for the electrostatic q-capacity have attracted increasing attention [2][3][4][5][6][7][8][9][10]. The electrostatic q-capacity also has applications in analysis, mathematical physics, and partial differential equations (see [11][12][13]).
The electrostatic q-capacity can be extended on function spaces. Let CðS n−1 Þ denote the set of continuous functions defined on S n−1 , which is equipped with the metric induced by the maximal norm. Write C + ðS n−1 Þ for the set of strictly positive functions in CðS n−1 Þ. For 1 < q < n and f ∈ C + ðS n−1 Þ, define the electrostatic q-capacity C q ð f Þ by where ½ f denotes the Aleksandrov body (also known as the Wulff shape) associated with f . For nonnegative f ∈ CðS n−1 Þ, the Aleksandrov body ½ f is defined by Obviously, ½ f is a compact convex set containing the origin and h ½ f ≤ f , where h ½ f denotes the support function of ½ f . Moreover, for every compact convex set K containing the origin. If f ∈ C + ðS n−1 Þ, then½ f is a convex body in R n containing the origin in its interior. The Aleksandrov convergence lemma reads: if the where the L p scalar multiplication a ⋅ f is defined by a a/p f . By the definition of the Aleksandrov body (3), we have for every u ∈ S n−1 , which was defined by Firey [14]. In the mid 1990s, it was shown in [15,16] that when L p Minkowski sum is combined with volume the result is an embryonic L p -Brunn-Minkowski theory. Zou and Xiong ([7], Theorem 3.11) established the functional form of the L p Brunn-Minkowski inequality for the electrostatic q-capacity. Suppose 1 < p < ∞ and 1 < q < n.
with equality if and only if ½ f and ½g are dilates. The Orlicz Brunn-Minkowski theory which was launched by Lutwak et al. in a series of papers [17][18][19] is an extension of the L p 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 : ½0,∞Þ ⟶ ½0,∞Þ with ϕð0Þ = 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 R n , then, the Orlicz sum a ⋅ K+ ϕ b ⋅ L is the convex body defined by for every u ∈ S n−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 L p Minkowski sum of functions to the Orlicz sum. For ϕ ∈ Φ, f , g ∈ C + ðS n−1 Þ, and a, b ≥ 0 (not both zero), the Orlicz sum a ⋅ f + ϕ b ⋅ g is defined by If we take ϕðtÞ = t p ðp ≥ 1Þ in (9), then it, induces the L p Minkowski sum (5). By the definition of the Aleksandrov body (3), (8), (9), and (4), we have ½a ⋅ h K + ϕ b ⋅ h L = a ⋅ K+ ϕ b ⋅ 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 electrostatic q-capacity.
If ϕ is strictly convex, equality holds if and only if ½ f and ½g are dilates.

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 in R n equipped with the standard Euclidean norm. Let x ⋅ y denote the standard inner product of x, y ∈ R n . For x ∈ R n , jxj = ffiffiffiffiffiffiffiffi x ⋅ x p denotes the Euclidean norm of x. We write B = fx ∈ R n : jxj ≤ 1g and S n−1 for the standard unit ball of R n and its surface, respectively. Each compact convex set K is uniquely determined by its support function h K : R n ⟶ R, which is defined by h K ðxÞ = max fx ⋅ y : y ∈ R n g, for x ∈ R n . Obviously, the support function is positively homogeneous of order 1.
The class of compact convex sets in R n is often equipped with the Hausdorff metric δ H , which is defined for compact convex sets K and L by Denote by K n the set of convex bodies in R n and by K n o the set of convex bodies which contain the origin in their interiors. For s > 0, the set sK = fsx : x ∈ Kg 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 ∈ R n . Let K, L ∈ K n , the Minkowski sum of K and L is the convex body Some properties of the electrostatic q-capacitary measure are required [2,3,7,8,11]. The electrostatic q-capacitary measure, μ q ðE, ⋅Þ, of a bounded open convex set E in R n is the measure on the unit sphere S n−1 defined for ω ⊂ S n−1 and 1 < q < n by where g −1 : S n−1 ⟶ ∂E (the set of boundary points of E) denotes the inverse Gauss map, H n−1 the ðn − 1Þ-dimensional 2
CNSXYZ [2] showed the Hadamard variational formula for the electrostatic q-capacity: for K, L ∈ K n and 1 < q < n, And variational formula (14) leads to the following Poincare q-capacity formula: The electrostatic q-capacity C q ðKÞ has the following properties. First, it is increasing with respect to set inclusion; that is, if K 1 ⊆ K 2 , then C q ðK 1 Þ ≤ C q ðK 2 Þ. Second, it is positively homogeneous of degree ðn − qÞ, i.e., C q ðsKÞ = s n−q C q ðKÞ for s > 0. Third, it is a rigid motion invariant, i.e., C q ðφK + xÞ = C q ðKÞ for x ∈ R n and φ ∈ OðnÞ. If q = 2, then (15) induces the Poincare capacity formula Let CðS n−1 Þ denote the set of continuous functions defined on S n−1 , which is equipped with the metric induced by the maximal norm. Write C + ðS n−1 Þ for the set of strictly positive functions in CðS n−1 Þ.
The Hadamard variational formula for the electrostatic q-capacity [2] states the following: For f ∈ C + ðS n−1 Þ, define Obviously, C q ðh K Þ = C q ðKÞ for every K ∈ K n o . By the Aleksandrov convergence lemma and the continuity of C q on K n , the functional C q : C + ðS n−1 Þ ⟶ ð0,∞Þ is continuous. For K ∈ K n o and g ∈ CðS n−1 Þ, the mixed elec-trostatic q-capacity C q ðK, gÞ is defined by Applying the Hadamard variational formula for the electrostatic q-capacity, the mixed electrostatic q-capacity C q ðK, gÞ has the following integral representation: Let L ∈ K n . If g = h L , then, C q ðK, gÞ is the mixed electrostatic q-capacity C q ðK, LÞ, which has the following integral representation: The Minkowski inequality for the electrostatic q-capacity ([2], Theorem 3.6) states the following: let1 < q < n.
If K, L ∈ K n , then, with equality if and only if K and L are homothetic. Let 1 ≤ p < ∞ and 1 < q < n. For K ∈ K n o and g ∈ CðS n−1 Þ, the L p Hadamard variational formula for the electrostatic q-capacity [7] states the following: The L p mixed electrostatic q-capacity C p,q ðK, gÞ is defined by Take g = h K in (24), and combine C q ðK, gÞ = C q ðKÞ to obtain the Poincare q-capacity formula (15). Zou and Xiong ([7], Theorem 3.9) established the L p Minkowski inequality for the L p electrostatic q-capacity: let 1 < p < ∞ and 1 < q < n. If K ∈ K n o and g ∈ CðS n−1 Þ, then, with equality if and only if K and ½g are dilates. Based on the Orlicz sum (9), we define the Orlicz mixed electrostatic q-capacity as follows. For K ∈ K n o and g ∈ CðS n−1 Þ, the Orlicz mixed electrostatic q-capacity C ϕ,q ðK, gÞ is defined by 3 Journal of Function Spaces Indeed, the Orlicz mixed electrostatic q-capacity can be extended on function spaces. Let ϕ ∈ Φ and 1 < q < n. For f ∈ C + ðS n−1 Þ and g ∈ CðS n−1 Þ, the Orlicz mixed electrostatic q-capacity C ϕ,q ð½ f , gÞ is defined by If f = h K with K ∈ K n o , then, C ϕ,q ð½h K , gÞ = C ϕ,q ðK, gÞ.

Main Results
The following variational formula of electrostatic q-capacity plays a crucial role in our proof.
for every u ∈ S n−1 .
The continuity properties of the Orlicz sum were established by Xi et al. [21].
(i) Let ff i g, fg i g ⊂ C + ðS n−1 Þ and fg i g ⊂ CðS n−1 Þ such that f i ⟶ f and g i ⟶ g, respectively. Then Then Due to Lemma 2, the integral representation of the Orlicz mixed electrostatic q-capacity is given.
Indeed, (36) can be considered as the Orlicz Hadamard variational formula for the electrostatic q-capacity. If we take ϕðtÞ = t p ð1 ≤ p<∞Þ and f = h K with K ∈ K n o in (36), then, we obtain the L p Hadamard variational formula (23).
Note that ½h K = K for every K ∈ K n o . Take f = h K in Lemma 4 to get Lemma 5. Suppose ϕ ∈ Φ and 1 < q < n. If K ∈ K n o and g ∈ C ðS n−1 Þ, then,

Journal of Function Spaces
A direct consequence of Lemma 4 and the homogeneity of the electrostatic q-capacitary measure can be obtained. Corollary 6. Suppose ϕ ∈ Φ and 1 < q < n. If f ∈ C + ðS n−1 Þ, then for every c > 0.
Let ϕ ∈ Φ, 1 < q < n, and K, L ∈ K n o . Note that K+ ϕ t ⋅ L = ½h K + ϕ t ⋅ h L , and apply (18) and (36) to obtain Based on (39), one can define the Orlicz mixed electrostatic q-capacity C ϕ,q ðK, LÞ of convex bodies K and L as follows: which was first defined by Hong et al. ( [10], Definition 3.1).
(ii) Let ff i g ⊂ C + ðS n−1 Þ and fg i g ⊂ CðS n−1 Þ such that f i ⟶ f and g i ⟶ g, respectively. Then, C ϕ,q ð½ f i , Applying Lemma 4, one concludes that (ii) holds.
Clearly, there exists a compact interval I ⊂ ð0,∞Þ such that g/f ∈ I for all u ∈ S n−1 .
(iii) directly follows from Lemma 4 and the fact that the sequence fϕ i ðtÞg converges uniformly to ϕðtÞ on I.
Next, we show that there is a natural Orlicz extension of the Minkowski inequality for the electrostatic q-capacity. Theorem 8. Suppose ϕ ∈ Φ and 1 < q < n. If f , g ∈ C + ðS n−1 Þ, then, If φ is strictly convex, then equality holds if and only if ½ f and ½g are dilates.
Proof. By the definition of the mixed electrostatic q-capacity (20) and the fact that h ½g ≤ g, we have for every f , g ∈ C + ðS n−1 Þ. From (31), Jensen's inequality, (20), (42), (22), and (18), it follows that It remains to prove the equality condition. Now, suppose ϕ 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 ∈ S n−1 with respect to the measure ð f ð⋅Þdμ q ð½ f , ⋅ ÞÞ/ðC q ð f ÞÞ. Then, we have where the last step is from the equality condition of (42). The definition of Aleksandrov body implies that h ½g = sh ½ f for almost every u ∈ S n−1 with respect to the measure ð f ð⋅Þdμ q ð½ f , ⋅ ÞÞ/ðC q ðf ÞÞ. Thus, for almost every u ∈ S n−1 with respect to the measure ð f ð⋅Þdμ q ð½ f , ⋅ ÞÞ/ðC q ðf ÞÞ. By the equality condition of the Minkowski inequality for the electrostatic q-capacity, there exists x ∈ R n such that ½g = s½ f + x.

Journal of Function Spaces
Hence, for almost every u ∈ S n−1 with respect to the measure f ð⋅Þdμ q ð½ f , ⋅ Þ, Since the centroid of μ q ð½ f , ⋅Þ is at the origin, we have that x ⋅ u = 0 for almost every u ∈ S n−1 with respect to the measure ð f ð⋅Þdμ q ð½ f , ⋅ ÞÞ/ðC q ð f ÞÞ. Note that the electrostatic q-capacitary measure μ q ð½ f , ⋅Þ is not concentrated on any great subsphere of S n−1 . Hence, x = 0, which in turn implies that ½ f and ½g are dilates.
Conversely, assume that ½ f and ½g are dilates, say, ½ f = c½g for some c > 0. From our assumption, Corollary 6, (18), and the fact that C q ðc½gÞ = c n−q C q ð½gÞ, it follows that This completes the proof.
By using the Orlicz-Minkowski inequality for the electrostatic q-capacity, we establish the following Orlicz Brunn-Minkowski inequality for the electrostatic q-capacity which is the general version of Theorem 1.

Theorem 9.
Suppose ϕ ∈ Φ and 1 < q < n. If f , g ∈ C + ðS n−1 Þ and a, b ≥ 0 (not both zero); then, If ϕ is strictly convex, then equality holds if and only if ½ f and ½g are dilates.
Proof. By (31), (9), and the Orlicz-Minkowski inequality for the electrostatic q-capacity (41), we have By the equality condition of the Orlicz-Minkowski inequality for the electrostatic q-capacity, we have that if ϕ is strictly convex, then equality in (48) holds if and only if ½ f and ½g are dilates of ½a ⋅ f + φ b ⋅ g. Remark 1. The case ϕðtÞ = t p ð1 ≤ p<∞Þ of Theorem 9 was established by Zou and Xiong [7].
For K, L ∈ K n o , take f = h K and g = h L in Theorem 9 to obtain the following Orlicz-Brunn-Minkowski inequality for the electrostatic q-capacity, which was established by Hong et al. [10].
Remark 2. The case φðtÞ = 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].
Proof. For t ≥ 0 and f , g ∈ C + ðS n−1 Þ, define the function GðtÞ by The Orlicz-Brunn-Minkowski inequality for the electrostatic q-capacity implies that GðtÞ is nonnegative. Obviously, Journal of Function Spaces On the other hand, by (51) and the continuity of C q , we have The continuity of C q and (27) imply From (53), (54), (55), and (52), it follows that 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 electrostatic q-capacity. Lemma 12. Suppose ϕ ∈ Φ and 1 < q < n. If f , g ∈ C + ðS n−1 Þ and C is a subset of C + ðS n−1 Þ such that f , g ∈ C, then the following assertions hold: (i) C ϕ,q ð½h, f Þ = C ϕ,q ð½h, gÞ for all h ∈ C; then ½ f = ½g (ii) ðC ϕ,q ð½ f , hÞÞ/ðC q ð f ÞÞ = ðC ϕ,q ð½g, hÞÞ/ðC q ð f ÞÞ for all h ∈ C; then ½ f = ½g Proof. We first show that (i) holds. Since C ϕ,q ð½ f , f Þ = ϕð1Þ C q ð f Þ, it follows that ϕð1Þ = ðC ϕ,q ð½ f , gÞÞ/ðC q ðgÞÞ by the assumption. By the Orlicz-Minkowski inequality for the electrostatic q-capacity, we have ϕð1Þ ≥ ϕðððC q ð f ÞÞ/ðC q ðgÞÞÞ 1/n−q Þ. The monotonicity of ϕ and 1 < q < n imply that with equality if and only if ½ f and ½g are dilates. This inequality is reversed if interchanging f and g. So,C q ð f Þ = C q ðgÞ and ½ f and ½g are dilates. Assume that s½ f = s½g for some s > 0. The homogeneity of C q implies s = 1. Thus, ½ f = ½g. Then, we can prove (ii) with the similar arguments in (i).
If the Orlicz mixed electrostatic q-capacity C ϕ,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].