Orlicz mixed chord integrals

We introduce a affine geometric quantity and call it Orlicz mixed chord integral, which generalize the chord integrals to Orlicz space. Minkoswki and Brunn-Minkowski inequalities for the Orlicz mixed chord integrals are establish. These new inequalities in special cases yield some isoperimetric inequalities for the usual chord integrals. The related concepts and inequalities of Lp-mixed chord integrals are also derived.

For K ∈ S n and u ∈ S n−1 , the half chord of K in the direction u, defined by.
If there exist constants λ > 0 such that d(K, u) = λd(L, u), for all u ∈ S n−1 , then star bodies K, L are said to have similar chord (see Gardner [1] or Schneider [26]). Lu [27] introduced the chord integral of star bodies: For K ∈ S n and 0 ≤ i < n, the chord integral of K, denoted by B i (K), defined by The main aim of the present article is to generalize the chord integrals to Orlicz the space. We In Section 3, we introduce a notion of Orlicz chord addition K+ φ L of star bodies K, L, defined by Here φ ∈ Φ 2 , the set of convex function φ : [0, ∞) 2 → (0, ∞) that are decreasing in each variable and satisfy φ(0, 0) = ∞ and φ(∞, 1) = φ(1, ∞) = 1. The particular instance of interest corresponds to using (1.5) with φ(x 1 , x 2 ) = φ 1 (x 1 ) + εφ 2 (x 2 ) for ε > 0 and some φ 1 , φ 2 ∈ Φ, where the sets of convex functions In accordance with the spirit of Aleksandrov [28], Fenchel and Jensen [29] introduction of mixed quermassintegrals, and introduction of Lutwak's [30] L p -mixed quermassintegrals, we are based on the study of first order variational of the chord integrals. In Section 4, we prove that the first order Orlicz variation of the chord integrals can be expressed as: For K, L ∈ S n , φ 1 , φ 2 ∈ Φ, 0 ≤ i < n and ε > 0, where φ ′ r (1) denotes the value of the right derivative of convex function φ at point 1. In this first order variational equation (1.4), we find a new geometric quantity. Based on this, we extract the required geometric quantity, denotes by B φ,i (K, L) and call as Orlicz mixed chord integrals, defined by We also show the new affine geometric quantity has an integral representation.
In Section 5, as application, we establish an Orlicz Minkowski inequality for the Orlicz mixed chord integrals: If K, L ∈ S n , 0 ≤ i < n and φ ∈ Φ, then . (1.7) If φ is strictly convex, equality holds if and only if K and L have similar chord. In Section 6, we establish an Orlicz Brunn-Minkowski inequality for the Orlicz chord addition and the chord integrals. If K, L ∈ S n , 0 ≤ i < n and φ ∈ Φ 2 , then . (1.8) If φ is strictly convex, equality holds if and only if K and L have similar chord.

Preliminaries
The setting for this paper is n-dimensional Euclidean space R n . A body in R n is a compact set equal to the closure of its interior. For a compact set K ⊂ R n , we write V (K) for the (n-dimensional) Lebesgue measure of K and call this the volume of K. Associated with a compact subset K of R n , which is starshaped with respect to the origin and contains the origin, its radial function is ρ(K, ·) : Note that the class (star sets) is closed under unions, intersection, and intersection with subspace. The radial function is homogeneous of degree −1, that is (see e.g. [1]), for all u ∈ S n−1 and r > 0. Letδ denote the radial Hausdorff metric, as follows, if K, L ∈ S n , theñ From the definition of the radial function, it follows immediately that for A ∈ GL(n) the radial function of the image AK = {Ay : y ∈ K} of K is given by (see e.g. [26]) for all x ∈ R n .
For K i ∈ S n , i = 1, . . . , m, define the real numbers R Ki and r Ki by obviously, 0 < r Ki < R Ki , for all K i ∈ S n , and writing R = max{R Ki } and r = min{r Ki }, where i = 1, . . . , m.

Mixed chord integrals
If K 1 , . . . , K n ∈ S n , the mixed chord integral of K 1 , . . . , K n , denotes by B(K 1 , . . . , K n ), defined by (see [27]) is written as B i (K) and call chord integral of K. Obviously, For K ∈ S n and 0 ≤ i < n, we have K, L ∈ S n and 0 ≤ i < n, it is easy that This integral representation (2.4), together with the Hölder inequality, immediately gives: The Minkowski inequality for the i-th mixed chord integral. If K, L ∈ S n and 0 ≤ i < n, then with equality if and only if K and L have similar chord.

L p -mixed chord integrals
and p ≥ 1 in (1.5), the Orlicz chord addition+ φ becomes a new addition+ p in L p -space, and call as L p -chord addition of star bodies K and L.
for u ∈ S n−1 . The following result follows immediately form (2.6) with p ≥ 1.
Definition 2.1 Let K, L ∈ S n , 0 ≤ i < n and p ≥ 1, the L p -chord integral of star K and L, denotes by B −p,i (K, L), defined by Obviously, when K = L, the L p -mixed chord integral B −p,i (K, K) becomes the chord integral B i (K).
This integral representation (2.7), together with the Hölder inequality, immediately gives:

8)
with equality if and only if K and L have similar chord. Take K+ p L for Q, recall that B p,i (Q, Q) = B i (Q), inequality (2.9) follows easy.

Orlicz chord addition
Throughout the paper, the standard orthonormal basis for R n will be {e 1 , . . . , e n }. Let Φ n , n ∈ N,  for u ∈ S n−1 . Equivalently, the Orlicz chord addition+ φ (K 1 , . . . , K m ) can be defined implicitly by for all u ∈ S n−1 .
if and only if Proof This follows immediately from definition 3.1. .
Proof This follows immediately from Lemma 3.4. Proof This follows immediately from Lemma 3.5.
Next, we define the Orlicz chord linear combination on the case m = 2.

Orlicz mixed chord integrals
In order to define Orlicz mixed chord integrals, we need the following Lemmas 4.1-4.4.
Proof This follows immediately from (3.4) and Lemma 3.5.
Denoting by B φ,i (K, L), for any φ ∈ Φ and 0 ≤ i < n, the integral on the right-hand side of (4.4) with φ 2 replaced by φ, we see that either side of the equation   Lemma 4.6 If φ 1 , φ 2 ∈ Φ, 0 ≤ i < n and K, L ∈ S n , then Lemma 4.7 If K, L ∈ S n , φ ∈ C and any A ∈ SL(n), then for ε > 0 Proof This follows immediately from (2.1) and (3.1).
We easy find that B φ,i (K, L) is invariant under simultaneous unimodular centro-affine transformation.
Lemma 4.8 If φ ∈ Φ, 0 ≤ i < n and K, L ∈ S n , then for A ∈ SL(n), Proof This follows immediately from Lemmas 3.3 and 4.7.

Orlicz chord Minkowski inequality
In this section, we need define a Borel measure in S n−1 , denotes by B n,i (K, υ), call as chord measure of star body K. .

(5.3)
If φ is strictly convex, equality holds if and only if K and L have similar chord.

(5.4)
If φ is strictly convex, equality holds if and only if K and L have similar chord.
Corollary 5.5 If K, L ∈ S n , 0 ≤ i < n and p ≥ 1, then with equality if and only if K and L are dilates.
Taking i = 0 in (5.6), this yields L p -Minkowski inequality is following: If K, L ∈ S n and p ≥ 1, then with equality if and only if K and L have similar chord.