SOME NEW BRUNN-MINKOWSKI-TYPE INEQUALITIES IN CONVEX BODIES

We establish some analogues of the Brunn-Minkowski inequalities on convex bodies and the Minkowski inequality and their inverse versions. As an application, we generalize and improve some interrelated results.


Introduction
The Brunn-Minkowski inequality theory plays an important role in a number of mathematical disciplines such as measure theory, crystallography, optimal control theory, functional analysis, and geometric convexity.It has many useful applications in combinatorics, stochastic geometry, and mathematical economics.
Notation and preliminary works are given in Section 2. In Section 3.1, some new inequalities of width-integrals, with equality conditions for convex bodies and their inverse versions are established.And we also establish some similar inequalities for projection bodies.This work generalizes a result which was given in [13].Most importantly, we find the Brunn-Minkowski inequality for polars of mixed projection bodies in the section.
Generalizations and inverse versions of the dual Minkowski inequality and Brunn-Minkowski inequalities, with inequality conditions, for the radial Minkowski linear combination are presented in Section 3.2.These results generalize some results which were given by Lutwak [12,17].
A general dual Brunn-Minkowski inequality, with equality conditions for radial Blaschke linear combination and its inverse version were given in Section 3.3.A special case of the result is just a new inequality, which was given by Lutwak [17].A dual Brunn-Minkowski inequality for the harmonic Blaschke linear combination and its inverse version, with equality conditions, were established in Section 3.4.896 Some new Brunn-Minkowski-type inequalities in convex bodies A general Minkowski inequality and Brunn-Minkowski inequalities of affine surface area for the Blaschke linear combination and their inverse versions (with equality conditions) for convex bodies are presented in Section 3.5.In this section, we also establish some similar inequalities for the harmonic linear combination.Our work generalizes some results of Lutwak [16,18].
Finally, in Section 3.6, we give the dual Brunn-Minkowski inequality of affine surface area and its inverse version, with equality conditions for the Lutwak linear combination.

Notations and preliminary ideas
The setting for this paper is n-dimensional Euclidean space R n (n > 2).Let C n denote the set of nonempty convex figures (compact, convex subsets) and let n denote the subset of C n consisting of all convex bodies (compact, convex subsets with nonempty interiors) in R n , and if p ∈ n , let n p denote the subset of n that contains the centered (centrally symmetric with respect to p) bodies.We reserve the letter u for unit vectors, and the letter B is reserved for the unit ball centered at the origin.The surface of B is S n−1 .For u ∈ S n−1 , let E u denote the hyperplane through the origin, that is orthogonal to u.We will use K u to denote the image of K under an orthogonal projection onto the hyperplane E u .

Mixed volumes and radial functions. We use
where u • x denotes the usual inner product u and x in R n .Let δ denote the Hausdorff metric on n ; that is, for where | • | ∞ denotes the sup-norm on the space of continuous functions C(S n−1 ).For a convex body K and a nonnegative scalar λ, λK is used to denote {λx : x ∈ K}.
where the sum is taken over all n-tuples (i 1 ,...,i n ) of positive integers not exceeding r.
The coefficient V i1,...,in depends only on the bodies K i1 ,...,K in , and is uniquely determined by (2.5), it is called the mixed volume of K i ,...,K in , and is written as ) is the ith projection measure (Quermassintegral) of K and is written as W i (K).With this notation, W 0 =V (K), while nW 1 (K) is the surface area of K, S(K).Associated with a compact subset K of R n , which is star-shaped with respect to the origin, its radial function ρ(K,•) : (2.6) If ρ(K,•) is positive and continuous, K will be called a star body.Let ϕ n denote the set of star bodies in R n and let ϕ n c denote the subset of ϕ n that contains the centered bodies (centrally symmetric with respect to c).If, for K ∈ ϕ n , ρ(K,•) is even (assumes the same values at antipodal points), then K is called centered.
The radial Hausdorff distance δ(K,L) of K,L ∈ ϕ n , is defined by Lutwak [18] as (2.7) It will be helpful to extend the domain of the radial function from S n−1 to R n − {0}.For K ∈ ϕ n , and The polar coordinate formula for the volume in R n is (2.9)

Width-integrals of convex bodies.
For u ∈ S n−1 , b(K,u) is defined to be half the width of K in the direction u.Two convex bodies K and L are said to have similar width if there exists a constant λ > 0 such that b(K,u) = λb(L,u) for all u ∈ S n−1 .For K ∈ n and p ∈ intK, we use K p to denote the polar reciprocal of K with respect to the unit sphere centered at p. Width-integrals were first considered by Blaschke [4,5].The width-integral of index i is defined by where dS is the (n − 1)-dimensional volume element on S n−1 .
898 Some new Brunn-Minkowski-type inequalities in convex bodies The width-integral of index i is a map It is positive, continuous, homogeneous of degree n − i, and invariant under motion.In addition, for i ≤ n, it is also bounded and monotone under set inclusion.
The following results due to Lutwak [13] will be used later: with equality if and only if K is symmetric with respect to p.

The radial Minkowski linear combination and dual mixed volumes.
We define a vector addition on R n , which we call radial addition as follows: if x 1 ,...,x r ∈ R n , then x 1 + ••• +x r is defined to be the usual vector sum of x 1 ,...,x r provided x 1 ,...,x r all lie in 1-dimensional subspace of R n , and as the zero vector otherwise.
If K 1 ,...,K r ∈ ϕ n and λ 1 ,...,λ r ∈ R, then the radial Minkowski linear combination Lutwak [17] as ( It has the following important property: for K,L ∈ ϕ n and λ,µ ≥ 0, For K 1 ,...,K r ∈ ϕ n and λ 1 ,...,λ r ≥ 0, the volume of the radial Minkowski linear combination where the sum is taken over all n-tuples (i 1 ,...,i n ) whose entries are positive integers not exceeding r.If we require the coefficients of the polynomial in (2.16) to be symmetric in their argument, then they are uniquely determined.The coefficient V i1,...,in is nonnegative and depends only on the bodies K i1 ,...,K in .It is written as For K i ∈ ϕ n , the dual mixed volumes are defined first by Lutwak [18] as (2.17) For K,L ∈ ϕ n and i ∈ R, the ith dual mixed volume of K and L, V i (K,L) is defined by 2.4.The radial Blaschke linear combination.Lutwak [17] defined the radial Blaschke linear combination.If K,L ∈ ϕ n and λ,µ ≥ 0, then λ • K +µ • L, as the star body whose radial function is given by (2.20) We will call the addition and scalar multiplication the radial Blaschke addition and scalar multiplication.We note that for convex bodies, Blaschke scalar multiplication and the radial Blaschke scalar multiplication agree.

The harmonic Blaschke linear combination.
A new addition called the harmonic Blaschke addition is defined by Lutwak [18].Suppose K,L ∈ ϕ n , and λ,µ ≥ 0 (not both zero).To define the harmonic Blaschke linear combination λK +µL, first define ξ > 0 by The body λK +µL ∈ ϕ n is defined as the body whose radial function is given by Lutwak [18] in the form From this definition and (2.9), it follows immediately that ξ = V (λK +µl), and hence (2.23)

Curvature functions and affine surface area.
A convex body K is said to have a positive continuous curvature function (see Lutwak [18]), for each The subset of n consisting of bodies which have a positive continuous curvature function will be denoted by κ n .The following result is true (see Lutwak [16]): for 900 Some new Brunn-Minkowski-type inequalities in convex bodies Suppose K,L ∈ κ n and λ,µ ≥ 0 (not both zero).From (2.26), it follows that the function λ f (K,•) + µ f (L,•) satisfies the hypothesis of Minkowski's existence theorem (see Bonnesen and Fenchel [6]).The solution of the Minkowski problem for this function is denoted by λ where the linear combination λK +µL is called the Blaschke linear combination.
The relationship between the Blaschke and Minkowski scalar multiplications is given by (2.28) The affine surface area of K ∈ κ n , Ω(K), is defined by It is well known that this functional is invariant under unimodular affine transformations.For K,L ∈ κ n , and i ∈ R, the ith-mixed affine surface area of K and L, Ω i (K,L), was defined by Lutwak [18] as (2.30) Now, we define the ith affine area of K ∈ κ n , Ω i (K), to be Ω i (K,B), since f (B,•) = 1, we have (2.31) Lutwak [17] also defined a mapping and pointed out that Λ transforms the harmonic Blaschke linear combination into the Blaschke linear combinations, that is, if K,L ∈ ϕ n c and λ,µ ≥ 0, then (2.34) Zhao Chang-Jian et al. 901 It can be shown that (see Lutwak [17]) for K,L,M ∈ n , where The class ν n is defined as follows: for some Q ∈ n .The class ν n has been extensively investigated by Petty [24].
Lutwak [18] again defined a new linear combination λ • K +µ • L ∈ ν n (in this paper it is called the Lutwak linear combination).For centrally symmetric K,L ∈ ν n , we have (2.37)

Inequalities for width-integrals
(1) The dual Brunn-Minkowski inequality for width-integrals of convex bodies.
For the projection measures of the Minkowski sum K + L of two convex bodies K and L, we have the following inequality: if K,L ∈ n , i < n − 1, with equality if and only if K and L are homothetic.
Similarly, for the width-integrals of the Minkowski sum of two convex bodies, Lutwak [13] established the following result with equality if and only if K and L have similar widths.Inequality (3.2) is called the dual Brunn-Minkowski inequality.
In fact, the inverse version of (3.2) also exists.
with equality if and only if K and L have similar widths.In addition, when i < n − 1, we also obtain that These inequalities are contained in the following theorem.
902 Some new Brunn-Minkowski-type inequalities in convex bodies The right equality of (3.5) holds if and only if K j have similar widths; the left equality of (3.5) holds if and only if, for almost all u, all but one of b(K j ,u) are zero.
with equality if and only if K j have similar widths.
In (3.6), let i = 2n; we have the following corollary.
with equality if and only if K and L have similar widths.
Taking i = 0 to (3.5), we get the following corollary.
Corollary 3.3.If K j ∈ n ( j = 1,...,m), then with equality if and only if K j have similar widths.
The following inequalities for width-integrals of the projection body will be estab- with equality if and only if Π 1 (M,K) and Π 1 (M,L) have similar widths.
with equality if and only if Π 1 (M,K) and Π 1 (M,L) have similar widths.These two inequalities are contained in the following theorem.
Proof.First, we give the proof of (3.11).From (2.10), (2.12), (2.35), and in view of the Minkowski inequality for integrals, we obtain On the other hand, we have The proof of (3.11) is complete.
904 Some new Brunn-Minkowski-type inequalities in convex bodies Similarly, we also can prove (3.12).Putting i = 2n and m = 2 to (3.12), we obtain the following corollary.
Corollary 3.5.If K,L,M ∈ n , then with equality if and only if Π 1 (M,K) and Π 1 (M,L) have similar widths.
From (3.15), (2.13), and in view of the centered projection body (centrally symmetric with respect to origin), we obtain the following corollary.
with equality if and only if Π 1 (M,K) and Π 1 (M,L) have similar widths.
This is just the Brunn-Minkowski inequality of polars of mixed projection bodies.This is a new result.

The dual Minkowski inequality and the Brunn-Minkowski inequality for the radial
Minkowski linear combination.The dual Minkowski inequality and the Aleksandrov-Fenchel inequality have been established by Lutwak [12].These results can be stated, respectively, as follows.
If K,L ∈ n , then with equality if and only if K is a dilation of L.
with equality if and only if K n−m+1 ,K n−m+2 ,...,K n are all dilations of each other.
In fact, the Brunn-Minkowski inequality can also be stated as (see Gardner [10], page 374) as follows.
If K,L ∈ n , then with equality if and only if K is a dilation of L.
The following general version of the dual Brunn-Minkowski inequality can be established.
with equality if and only if K is a dilation of L. This is contained in the following theorem.
The right equality of (3.21) holds if and only if K j are all dilations of each other; the left equality of (3.21) holds if and only if, for almost all u, all but one of ρ(K j ,u) are zero.
with equality if and only if K j are all dilations of each other.
Proof.We only give the proof of (3.21).From (2.15), (2.19), and in view of the Minkowski inequality for integrals, we obtain Wi K j 1/(n−i) . (3.23) On the other hand, we have Wi K j .906 Some new Brunn-Minkowski-type inequalities in convex bodies In (3.21), let i = 0; we have the following corollary.
Corollary 3.8.If K j ( j = 1,2,...,m) ∈ n , then The right equality of (3.25) holds if and only if K j are all dilations of each other; the left equality of (3.25) holds if and only if, for almost all u, all but one of ρ(K j ,u) are zero.
In (3.22), let i = 2n to obtain the following corollary.
Corollary 3.9.If K j ( j = 1,2,...,m) ∈ n , then The right equality of (3.26) holds if and only if K j are all dilations of each other; the left equality of (3.26) holds if and only if, for almost all u, all but one of ρ(K j ,u) are zero.
The dual Minkowski inequality (3.17) is contained in the following theorem.
with equality if and only if K is a dilation of L, while if i > n or i < 0, with equality if and only if K is a dilation of L.
To prove (3.27), we use (2.18) and in view of the Hölder inequality for integrals (see Hardy et al. [11], page 140) with Similarly, we also can prove (3.28).
An immediate consequence of Theorem 3.10 is the following theorem.
Theorem 3.11.If γ ⊂ ϕ n , K,L ∈ ϕ n , then i = 0 and i = n if either holds, it follows that K = L.
Zhao Chang-Jian et al. 907 Proof.We only prove the case of i > n.Suppose (3.29) holds.Take K for M, use (3.28), and in view of (2.9) and (2.17), we get with equality if and only if K and L are dilations.Now take L for M and get and K and L are dilations, the proof is complete.The same sort of argument shows that (3.30) leads to the same conclusion.
Remark 3.12.When n > i > 0 and let i = 1, Theorem 3.11 reduces to the following result.
holds, it follows that K = L.This is also a new result which was given by Lutwak [17].

The dual Brunn-Minkowski inequality for the radial Blaschke linear combination.
Lutwak [17] proved the following result.If K,L ∈ ϕ n , then with equality if and only if K is a dilation of L.
Our main result is given in the following theorem.

.36)
with equality if and only if K and L are dilations, while for n > i > 1 or i > n, with equality if and only if K j are homothetic.
A general version of the dual Minkowski inequality for the affine surface area will be given in the following theorem.
with equality if and only if K and L are homothetic.
An immediate consequence of Theorem 3.18 is the following theorem.Since Ω(K) = Ω(L) and K and L are homothetic, K and L must be translates.The same sort of argument shows that (3.51) leads to the same conclusion.
In Theorem 3.19, when i = −1; we have the following corollary.This is just a new result which was also given by Lutwak [18].Some new inequalities of the affine surface area for the harmonic Blaschke linear combination will be established.
Ω i ΛK j (n+1)/(n−i) . (3.59) The right equality of (3.59) holds if and only if K j are homothetic; the left equality of (3.59) holds if and only if, for almost all u, all but one of f (ΛK j ,u) are zero.

Theorem 3 . 21 .
If K j ( j = 1,...,m) ∈ ϕ n c , then, for i < −1, The right equality of(3.46)holds if and only if K j are homothetic; the left equality of(3.46)holds if and only if, for almost all u, all but one of ρ(K j ,u) are zero, while, for −1 < i < n or i > n, .37) with equality if and only if K and L are dilations.