L p Radial Blaschke-Minkowski Homomorphisms and L p Dual Affine Surface Areas

Schuster introduced the notion of radial Blaschke-Minkowski homomorphism and considered the Busemann-Petty problem for volume forms. Whereafter, Wang, Liu and He presented the Lp radial Blaschke-Minkowski homomorphisms and extended Schuster’s results. In this paper, associated with Lp dual affine surface areas, we give an affirmative and a negative form of the Busemann-Petty problem and establish two Brunn-Minkowski inequalities for the Lp radial Blaschke-Minkowski homomorphisms.


Introduction
If K is a compact star shaped (about the origin) in n-dimensional Euclidean space R n , then its radial function, ρ K = ρ(K, •) : R n \{0} → [0, ∞), is defined by (see [1]) If ρ(K, •) is positive and continuous, K will be called a star body (about the origin).The set of all star bodies in R n denotes by S n o .For the set of all origin-symmetric star bodies, we write S n os .Let S n−1 denote the unit sphere in R n .Two star bodies K and L are said to be dilated (of one another) if ρ K (u)/ρ L (u) is independent of u ∈ S n−1 .
Intersection bodies were explicitly defined and named by Lutwak (see [2]).For K ∈ S n o , the intersection body, IK, of K is a star body whose radial function is defined by for all u ∈ S n−1 .Here u ⊥ is the (n − 1)-dimensional hyperplane orthogonal to u and V n−1 denotes the (n − 1)-dimensional volume.
Problem 1 (Busemann-Petty problem).For K, L ∈ S n o , is there the implication Here V(K) denotes the n-dimensional volume of body K.
Remark 1. Problem 1 was stated by Lutwak (see [2]).If K, L ∈ S n os , then Problem 1 is called the symmetric Busemann-Petty problem.Gardner [13], Zhang [14] showed that the symmetric Busemann-Petty problem has an affirmative answer for n ≤ 4 and a negative answer for n ≥ 5.
For Problem 1, Lutwak [2] gave its an affirmative answer if K is restricted to the class of intersection bodies and two negative answers if K is not origin-symmetric star body or L is not intersection body.For more research on solutions to the Busemann-Petty problem, see e.g., [1,[13][14][15][16][17][18][19][20].
In 2006, based on the properties of intersection bodies, Schuster [21] introduced the radial Blaschke-Minkowski homomorphism, which is the more general intersection operator as follows: o and all ϕ ∈ SO(n).
Here " +" and " +" denote radial Blaschke addition and radial Minkowski addition, respectively; SO(n) denotes the group of rotation transformations.
Obviously, Problem 2 is a more general Busemann-Petty problem compared with Problem 1.For Problem 2, Schuster [22] gave an affirmative answer if K belongs to ΨS n o (the range of Ψ) and two negative forms.
In 2011, Wang, Liu and He [23] extended Schuster's radial Blaschke-Minkowski homomorphisms to L p analogies, and gave the notion of L p radial Blaschke-Minkowski homomorphisms as follows: o and all ϕ ∈ SO(n).
Here " + p " and " + p " denote L p radial Blaschke addition and L p radial Minkowski addition, respectively.
The L p dual affine surface areas firstly were introduced by Wang, Yuan and He (see [34]).Here, we improve Wang, Yuan and He's definition as follows: For K ∈ S n o and p > 0, the L p dual affine surface area, Ω p (K), of K is defined by Here V p (M, N) denotes the L p dual mixed volume of M, N ∈ S n o , and Q * denotes the polar of Q which is defined by (see [1]) If Q belongs to the set of convex bodies (or star bodies) whose centroid at the origin, then Equation ( 1) is just Wang, Yuan and He's (or Wang and Wang's) definition (see [34] or [35]).For the studies of L p dual affine surface areas, also see [36][37][38][39][40][41].
Remark 2. Recall that Lutwak's L p affine surface area was defined as follows (see [42]): For K ∈ K n o and p ≥ 1, the L p affine surface area, Ω p (K), of K is defined by Here, K n o denotes the set of convex bodies (compact, convex subsets with nonempty interiors) containing the origin in their interiors in R n and V p (M, N) is the L p mixed volume of M and N (see [42]).Compare to Equation (1) and Equation (2), we see that Equation ( 1) is really the duality of Equation (2).
In this paper, associated with L p dual affine surface areas, we research the L p radial Blaschke-Minkowski homomorphisms.We firstly consider the following Busemann-Petty problem of L p radial Blaschke-Minkowski homomorphisms.Problem 3. Let p > 0 and Ψ p : S n o → S n o be a L p radial Blaschke-Minkowski homomorphism.For K, L ∈ S n o , is there the implication For Problem 3, according to Equation (3), we obtain an affirmative form as follows: In addition, Furthermore, when K / ∈ S n os , by Equation ( 1) we give the following a negative form of Problem 3.
Next, associated with L q radial Minkowski sum and L q harmonic Blaschke sum of star bodies, we establish the following L p dual affine surface area forms of Brunn-Minkowski inequalities for the L p radial Blaschke-Minkowski homomorphisms, respectively.
with equality if and only if K and L are dilated.
Let q = n − p in Theorem 3 and notice that K + n−p L = K + p L (see Equation ( 7)), we obtain a Brunn-Minkowski inequality for the L p radial Blaschke sum K + p L.
with equality if and only if K and L are dilated.
with equality if and only if K and L are dilated.Here K ∓ q L denotes the L q harmonic Blaschke sum of K and L.
The proofs of Theorem 1 and Theorem 2 are completed in Section 3. In Section 4, we will give the proofs of Theorem 3 and Theorem 4.

General L p Radial Blaschke Bodies
For K, L ∈ S n o , real p = 0 and λ, µ ≥ 0 (not both 0), the L p radial Minkowski combination, o , of K and L is defined by (see [43,44]) Here " + p " denotes the L p radial Minkowski sum and λ In 2015, Wang and Wang [35] defined the L p radial Blaschke combinations of star bodies as follows: For K, L ∈ S n o , n > p > 0 and λ, µ ≥ 0 (not both 0), the L p radial Blaschke combination, Here " + p " denotes the L p radial Blaschke sum and λ From the definitions of above two combinations, we easily see In Equation ( 6), let with τ ∈ [−1, 1] and L = −K, and write We call ∇ τ p K the general L p radial Blaschke body of K. From Equations ( 8) and ( 9), we easily see that For the general L p radial Blaschke bodies, by Equation ( 8) we know that f 1 (τ) + f 2 (τ) = 1.This and Equation (9) give that if K ∈ S n os then ∇ τ p K ∈ S n os .If K / ∈ S n os , we have the following conclusion.
Theorem 5.For K, L ∈ S n o and p > 0.

L p Dual Mixed Volumes
Based on the L p radial Minkowski combinations of star bodies, a class of L p dual mixed volumes were introduced as follows (see [45,46]): For M, N ∈ S n o , p = 0 and ε > 0, the L p dual mixed volume, V p (M, N), of M and N is defined by From the above definition, L p dual mixed volume has the following integral representation (see [45,46]):

L q Harmonic Blaschke Sums
The harmonic Blaschke sums of star bodies were introduced by Lutwak (see [47]).For M, N ∈ S n o , the harmonic Blaschke sum, M ∓ N ∈ S n o , of M and N is defined by Based on above definition, Feng and Wang ( [48]) defined the L q harmonic Blaschke sums as follows: For M, N ∈ S n o , real q > −n, the L q harmonic Blaschke sum, M ∓ q N ∈ S n o , of M and N is given by

A Type of Busemann-Petty Problem
Theorems 1 and 2 show a type of Busemann-Petty Problem of the L p radial Blaschke-Minkowski homomorphisms for the L p dual affine surface areas.In this section, we will prove them.In order to prove Theorem 1, the following lemma is essential.

Lemma 1 ([23]
).If M, N ∈ S n o and p > 0, then Proof of Theorem 1.Since Ψ p K ⊆ Ψ p L, thus using Equation ( 12) we know that for p > 0 and any This together with Equation ( 14) yields Hence, by Equation (3) we have Obviously, we see that The proof of Theorem 2 needs the following lemmas.
According to the equality conditions of Equation ( 16), we see that equality holds in Equation ( 17) for τ ∈ (−1, 1) if and only if K and −K are dilated, i.e., K is origin-symmetric.For τ = ±1, Equation ( 17) becomes an equality.Lemma 4 ([23]).For p > 0, a map Ψ p : S n o → S n o is a L p radial Blaschke-Minkowski homomorphism if and only if there is a non-negative measure µ ∈ M(S n−1 , e) such that for K ∈ S n o , ρ(Ψ p K, •) p is the convolution of ρ(K, •) n−p and µ, namely Here e denotes the pole point of S n−1 and M(S n−1 , e) denotes the signed finite Borel measure space on S n−1 (see [21]).

Brunn-Minkowski Inequalities for the L p Radial Blaschke-Minkowski Homomorphisms
Associated with L q radial Minkowski sum and L q harmonic Blaschke sum of star bodies, Theorems 3 and 4 respectively give the L p dual affine surface area forms of Brunn-Minkowski inequalities for the L p radial Blaschke-Minkowski homomorphisms.In this section, we will complete their proofs.For the proof of Theorem 3, the following lemmas are essential.Lemma 6.If K, L ∈ S n o , p > 0 and 0 < q ≤ n − p, then for any u ∈ S n−1 , with equality for 0 < q < n − p if and only if K and L are dilated.For q = n − p, Equation ( 22) becomes an equality.
Proof.Because of 0 < q < n − p implies n−p q > 1, thus by Equation ( 19) and the Minkowski integral inequality we have for µ ∈ M(S n−1 , e) and any u ∈ S n−1 , This yields inequality Equation (22).
From the equality condition of Minkowski integral inequality, we know that equality holds in Equation (22) for 0 < q < n − p if and only if K and L are dilated.Clearly, if q = n − p, Equation ( 22) becomes an equality.Lemma 7. If K, L ∈ S n o , 0 < p < n/2 and 0 < q ≤ n − p, then for any M ∈ S n o , with equality if and only if K and L are dilated.
For 0 < q < n − p, according to the equality conditions of Equation ( 22) and Minkowski integral inequality, we see that equality holds in Equation ( 23) if and only if K and L are dilated, and Ψ p K and Ψ p L are dilated.However, by Equation ( 19) we know that K and L are dilated is equivalent to Ψ p K and Ψ p L are dilated.Therefore, equality holds in Equation ( 23) if and only if K and L are dilated.
For q = n − p, from equality condition of Minkowski integral inequality, we are aware of equality holds in Equation ( 23) if and only if Ψ p K and Ψ p L are dilated, this is equivalent to K and L are dilated.
Proof of Theorem 3.Because of pq (n−p) 2 > 0, thus by Equations ( 1) and ( 23), we obtain that From this, we get inequality Equation (4), and equality holds in Equation (4) if and only if K and L are dilated.
The proof of Theorem 1 requires the following lemmas.Lemma 8.If K, L ∈ S n o , 0 < p < −q < n, then for any u ∈ S n−1 , ρ p(n+q) n−p Ψ p (K∓ q L) (u) with equality if and only if K and L are dilated.
Proof.Since 0 < p < −q < n, thus n−p n+q > 1.Hence, by Equation ( 13), Equation (19) and the Minkowski integral inequality we have for µ ∈ M(S n−1 , e) and any u ∈ S n−1 , ρ p(n+q) n−p Ψ p (K∓ q L) (u) V(K ∓ q L) = ρ p Ψ p (K∓ q L) (u) This deduces Equation (24).According to the equality condition of Minkowski integral inequality, we know that equality holds in Equation (24) if and only if K and L are dilated.Lemma 9.If K, L ∈ S n o , 0 < p < n/2 and 0 < p < −q < n, then for any M ∈ S n o , V p (Ψ p (K ∓ q L), M) with equality if and only if K and L are dilated.

2 V
(K ∓ q L) = 1 n S n−1 ρ p(n+q) n−p Ψ p (K∓ q L) (u) V(K ∓ q L) (n−p) 2 p(n+q) ρ p M (u)du p(n+q) (n−p) 2 o denote the range of Ψ p and Ψ * p S n o denote the set of polars of all elements in Ψ p S n o , then Ψ * p S n o ⊆ S n o .From this, we write that