Some Inequalities about the General L p -Mixed Width-Integral of Convex Bodies

: The Brunn-Minkowski type and the cyclic Brunn-Minkowski type inequalities for the i-th general L p -mixed width-integral of convex bodies are established. Further, two cyclic in‐ equalities for the differences of i-th general L p -mixed width-integral of convex bodies are obtained


Introduction and Main Results
The setting for this paper is n -dimensional Euclidean spaces R n .Let K n denote the set of convex bodies (compact, convex subsets with non-empty interiors) in Euclidean space R n .K n o denotes the set of convex bodies containing the origin in their interiors.V (K) denotes the n-dimensional volume of a body K, B the standard unit ball, and V (B) = ω n .Let S n -1 denote the unit sphere in R n .
Blaschke [1] considered the classical width-integral of convex bodies first and Hadwiger [2] studied it further.In 1975 Lutwak [3] introduced the i-th width-integral of convex bodies.In 1977, Lutwak [4] generalized the i-th width-integral to the mixed width-integral of convex bodies.In 2016, Feng [5] gave the definitions of mixed width-integral and the general i-th width-integral of convex bodies.In 2017, Zhou [6] defined the general L pmixed width-integral of convex bodies.For the more results of the mixed width-integral of convex bodies, we refer the interested reader to Refs.[7-14].
In this paper, we first establish the Brunn-Minkowski type inequality for the i-th general L p -mixed width-integral of convex bodies.
the equality holds in (1) or (2) if and only if K and L have similar general L p -width, and (B (τ) pi (M )B (τ) pi (K)) = c(B (τ) pi (N)B (τ) pi (L)) , where c is a constant.We also establish two cyclic inequalities for the differences of i-th general L p -mixed width-integral of convex bodies.
with equality if and only if K has constant general L p -width.
with equality if and only if (b (τ) , where c is a constant, and K and L have similar general L p -width.
Meanwhile, we establish a cyclic Brunn-Minkowski inequality for the i-th general L p -mixed width-integral of convex bodies.(5)   with equality if and only if K and L both have constant general L p -width.If n -p < j < n and j ≤ i < k, or j > n and i ≤ j < k, the inequality is reversed.

Support Function and Firey L p -Combination
If K Î K n , the support function, h K = h(K ×): R n ®(-¥¥), is defined by [15,16] h(Kx) = max{x × y:y Î K}x Î R n where x × y denotes the standard inner product of x and y.
For KL Î K n o , p ≥ 1 and λ, μ ≥ 0 (not both zero), the Firey L p -combination λ × K + p μ × L Î K n o of K and L is defined by [17] h(λ where the operation "+ p " is called Firey addition and λ × K denotes the Firey scalar multiplication.

General L p -Mixed Width-Integral of Order i
For τ Î[-11] and p > 0, the general L p -mixed width-integral where b (τ) 1 p for any u Î S n -1 , and f 1 (τ), f 2 (τ) are chosen as follows: K and L are said to have similar general L p -width if there exists a constant λ > 0 such that b (τ) p (Lu) for all u Î S n -1 , then we call K and L have the same general L p -width.If b (τ) p (Ku) is a constant for all u Î S n -1 , we call K has the constant general L p -width. Taking Further, let L = B in (8), since b (τ) p (Bu) = 1, and write B (τ) pi (K) for B (τ) pi (KB), we get where

Proofs of Theorems
In this section, we give the proofs of the Theorems 1-4.The proof of Theorem 1 requires the following lemmas.
Lemma 1 [6] If KL Î K n o , τ Î[-11] and p > 0, for i ≤ n -p ≤ j ≤ n and i ¹ j, then (11)   with equality in every inequality if and only if K and L have similar general L p -width.
Lemma 2 [18,19] Let x = (x 1 x n ) and y = (y 1 y n ) be two series of non-negative real numbers, and for p < 0 or 0 < p < 1, with equality in every inequality if and only if x and y are proportional.
Proof of Theorem 1 Let j = n in Lemma 1, since B (τ) pn (K + p L) = ω n is a constant, for i ≤ n -p, we immediately obtain the equality holds if and only if K and L have similar general L p -width, which is just Corollary 3.
Since M, N, K, LÎ K n o , M and N have similar general L p -width, for n -p < i < n, we have By the definition of the i-th general L p -mixed width-integral of convex bodies, we know that B (τ) pi (K)≤ B (τ) pi (M ) and According to (13), combining ( 16) with (17), we can obtain the equality holds if and only if (B (τ) pi (M )B (τ) pi (K)) is proportional to (B (τ) pi (N)B (τ) pi (L)), and K and L have similar general L p -width.The proof of inequality ( 2) is similar.This proves the theorem.
The proof of Theorem 2-3 requires the following lemma.
Lemma 3 [20] Suppose that f i , g i (i= 1, 2) are non-negative continuous functions on S n -1 such that ∫ , and for all ξ Î S n -1 , f p 1 (ξ) = λg q 1 (ξ) where λ is a constant, then ) with equality if and only if f p 2 (ξ) = λg q 2 (ξ) for any ξ Î S n -1 .Proof of Theorem 2 Suppose that 0 , that means K has constant general L p -width.This proves the theorem.
Proof of Theorem 3 We can prove Theorem 3 by Lemma 3 as well.Suppose that 0 ≤ i < j < k ≤ nλ = (k -i)/( j -i) and μ = (k -i)/(k -j), let By Lemma 3, the equality in (4) holds if and only if (b (τ) , that means K and L have similar general L p -width.This proves the theorem.
Taking i = 0, j = 1, k = n in Theorem 3, we obtain with equality if and only if (b (τ) , where c is a constant.Proof of Theorem 4 If j < n -p, combined ( 6) with ( 9), according to the Minkowskis inequality [21] , it follows that Since i < j < k means ki kj > 1, using Hölders inequality [22] we have Hence, we can get the following inequality Similarly, we can also obtain By ( 21), ( 23) and (24), we get From the equality condition of the Minkowskis inequality, we see that the equality (21) holds if and only if K and L have similar general L p -width.By the equality conditions of Hölders inequality, equality holds in (22) if and only if K has constant general L p -width.Similary, the equality holds in (24) if and only if L has constant general L p -width.Thus, the equality holds in (5) or its reverse if and only if both K and L have constant general L p -width.
Corollary 2 Let K Î K n o , p > 0, τ Î[-11], for i < j < k, then B (τ) pj (K) k -i ≤ B (τ) pi (K) k -j B (τ) pk (K) j -i (26) with equality if and only if K has constant general L p -width.Let i = j in Theorem 4, we may obtain the following Brunn-Minkowski inequality for the i-th general L p -mixed width-integral.
Corollary 3 For KL Î K n o , p > 0, τ Î[-11], i ¹ n, if i ≤ n -p, then B (τ) pi (K with equality if and only if K and L have simliar general L p -width.If n -p < i < n, or i > n, inequality (27) is reversed.
according to Lemma 3, we can get the inequality (3).By Lemma 3, the equality in (3) holds if and only if b