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In this paper, we establish the dual Orlicz-Minkowski inequality and the dual Orlicz-Brunn-Minkowski inequality for dual Orlicz mixed quermassintegrals.

Recently, Convex Geometry Analysis has made great achievement in Orlicz space (see [

Let K and L be two star bodies about the origin in

The case

Let

In this paper, we will define the dual Orlicz mixed quermassintegral

The main purpose of this paper is to establish the dual Orlicz-Minkowski inequality and the dual Orlicz-Brunn-Minkowski inequality for dual Orlicz mixed quermassintegrals.

Theorem 1.1 Let K and L be two star bodies about the origin in

with equality if and only if K and L are dilates of each other.

Theorem 1.2 Let K and L be two star bodies about the origin in

with equality if and only if K and L are dilates of each other.

This paper is organized as follows: In Section 2 we introduce above interrelated notations and their background materials. Section 3 contains the proofs of our main results.

The radial function

If

If

each other.

If

where the sum is taken over all n-tuples

where S is the Lebesgue measure on

The coefficients

gral

then

The dual mixed quermassintegral

where S is the Lebesgue measure on

By using the Minkowski’s integral inequality, we can obtain the dual Minkowski inequality for dual mixed quermassintegrals: If

equality holds if and only if K and L are dilates of each other.

Suppose that m is a probability measure on a space X and

If

Let

For

Corollary 3.1 The dual Orlicz mixed quermassintegral

Proof. Let

Lemma 3.1 [

if and only if

Lemma 3.2 [

uniformly for all

Theorem 3.1 Let

Proof. Suppose

uniformly on

Hence

We complete the proof of Theorem 3.1. ,

From (3.1) and Theorem 3.1, we have

For

ity measure on

Proof of Theorem 1.1

By (3.1), (2.6), (2.5) and the fact that

This gives the desired inequality. Since

Conversely, when

The following uniqueness is a direct consequence of the dual Orlicz-Minkowski inequality (1.4).

Corollary 3.2 Suppose

or

then

Proof. Suppose (3.4) holds. If we take K for M, then from (3.1), we obtain

Hence, from the dual Orlicz-Minkowski inequality (1.4), we have

with equality if and only if K and L are dilates of each other. Since

with equality if and only if K and L are dilates of each other. If we take L for M, we similarly have

Next, suppose (3.5) holds. If we take K for M, then from (3.1), we obtain

Then, from the dual Orlicz-Minkowski inequality (1.4), we have

with equality if and only if K and L are dilates of each other. Since

with equality if and only if K and L are dilates of each other. If we take L for M, we similarly have

From the dual Orlicz-Minkowski inequality, we will prove the following dual Orlicz-Brunn-Minkowski inequality which is more general than Theorem 1.2.

Theorem 3.2 Let

with equality if and only if K and L are dilates of each other.

Proof. Let

By the equality condition of the dual Orlicz-Minkowski inequality (1.4), equality in (3.6) holds if and only if K and L are dilates of each other.

Indeed, we also can prove the dual Orilcz-Minkowski inequality by the dual Orilcz- Brunn-Minkowski inequality.

Proof. For

is non-positive. Obviously,

On the other hand, we have

Let

By (3.3), we have

From (3.8), (3.9), and (3,10), it follows that

Combing (3.7) and (3.11), we have

Therefore, the equality in (3.12) holds if and only if

Remark 3.1 The case

Liu, L.J. (2016) Inequalities for Dual Orlicz Mixed Quermassintegrals. Advances in Pure Mathematics, 6, 894-902. http://dx.doi.org/10.4236/apm.2016.612067