Volume inequalities for -Minkowski combination of convex bodies
© Liu and Leng; licensee Springer 2013
Received: 29 November 2012
Accepted: 8 March 2013
Published: 27 March 2013
Recently Böröczky, Lutwak, Yang and Zhang proved the -Brunn-Minkowski inequality for two origin-symmetric convex bodies in the plane. This paper extends their results to m () origin-symmetric convex bodies in the plane. Moreover, relying on the recent results of Schuster and Weberndorfer, volume inequalities for -Minkowski combination of origin-symmetric convex bodies in and its dual form are established in this paper.
Keywords-Minkowski combination -Brunn-Minkowski inequality cone-volume probability measure normalized -mixed volume Wulff shape
The setting for this article is an Euclidean space , . A convex body is a compact convex subset of with a non-empty interior. For a compact convex set and , the support function is defined by , where denotes the standard inner product of x and y in . The polar body of a convex body K is given by . The Minkowski addition of two convex bodies K and L is defined as , and the scalar multiplication λK of K, where , is defined as .
with equality if and only if are equal.
Recently, Böröczky et al.  defined the -Minkowski combination of convex bodies and proved the -Brunn-Minkowski inequality, which is stronger than (1.1), for two origin-symmetric convex bodies in the plane.
Theorem 1.1 
When , equality in the inequality holds if and only if K and L are dilates or K and L are parallelograms with parallel sides.
Our first main result of this paper is to extend Theorem 1.1 to m () origin-symmetric convex bodies in the plane.
However, the -Brunn-Minkowski inequality in is still an open problem, even for origin-symmetric convex bodies.
Now, with our second main result we focus on the volume estimate for -Minkowski combination of origin-symmetric convex bodies in .
In , Schuster and Weberndorfer established two powerful volume inequalities of the Wulff shape determined by an f-centered isotropic measure ν (see Section 2 for details). Using their results, we establish the following two inequalities.
Furthermore, inequalities of mixed volume and normalized -mixed volume (given in this paper) for -Minkowski combination of not necessarily origin-symmetric convex bodies in are established in the following theorems.
where and are the surface area measure and the cone-volume measure of K, respectively (see Section 2 for the definitions).
where is the cone-volume probability measure of K (also see Section 2 for the definition).
Combining the famous variant (proved in ) of Aleksandrov’s lemma and the representation of (1.4), we obtain a limit form of in the following theorem.
The paper is organized as follows. In Section 2 some of the basic notations and preliminaries are provided. Section 3 contains the proofs of the main theorems. Some properties of normalized -mixed volume and Wulff shape are discussed in Section 4.
2 Notations and preliminaries
The group of nonsingular linear transformations is denoted by ; its members are, in particular, bijections of onto itself. The group of special linear transformations of is denoted by . These are the members of whose determinant is one.
For , let , and denote the transpose, inverse and inverse of the transpose of ϕ, respectively.
Recall that for a Borel set , the surface area measure of a convex body K in is the -dimensional Hausdorff measure of the set of all boundary points of K at which there exists a normal vector of K belonging to ω.
where is the mixed area measure of ().
where the notation signifies that K appears times and B appears i times.
where denotes the -dimensional Hausdorff measure.
Throughout, all Borel measures are understood to be non-negative and finite. We write suppν for the support of a measure ν.
where denotes the centroid of .
3 Proof of main results
The following lemma will be used in the proof of Theorem 1.2.
also contains the origin in its interior.
for all .
Hence, by (1.2), we have , which yields the lemma directly. □
Proof of Theorem 1.2 We will prove Theorem 1.2 by induction on m.
Obviously, it is true by Theorem 1.1 when .
We now consider the situation on m. In fact, since () are origin-symmetric convex bodies in the plane, then by (3.1) we have is also an origin-symmetric convex body in the plane.
Note that Theorem 1.2 also holds when and , respectively.
Thus Theorem 1.2 holds for all satisfying . □
In , Schuster and Weberndorfer established a sharp bound for the volume of the Wulff shape determined by an f-centered isotropic measure ν as follows.
Lemma 3.2 
with equality if and only if is a regular simplex inscribed in and f is constant on suppν.
A natural dual to Lemma 3.2 is also given in , which provided a sharp lower bound for the volume of the polar of the Wulff shape .
Lemma 3.3 
with equality if and only if is a regular simplex inscribed in and f is constant on suppν.
Proof of Theorem 1.4 Let and for all in Lemma 3.3. Similarly, from the proof of Theorem 1.3, we know that f is a positive continuous function on and ν is an isotropic f-centered measure.
Letting in Theorem 1.5 gives the following inequality of mixed volumes.
Letting in Theorem 1.5 gives the following inequality of quermassintegrals.
In view of (2.4), we also obtain the following inequality of mean widths.
The following variant of Aleksandrov’s lemma (see [, p.103]) will be needed in proving Theorem 1.7.
Lemma 3.7 
Observe that is the support function of K, hence .
4 Other results and comments
Firstly, we prove that -Minkowski combination and normalized -mixed volume are invariant under simultaneous unimodular centro-affine transformations.
The following proposition shows the property of weak convergence of the cone-volume probability measure.
Proposition 4.3 If is a sequence of convex bodies in that contain the origin in their interiors, and , where is a convex body that also contains the origin in its interior, then weakly.
The continuity of the normalized -mixed volume is contained in the following proposition.
Proposition 4.4 Suppose that and are two sequences of convex bodies in that contain the origin in their interiors, and , , where K and L are convex bodies that also contain the origin in its interior, then .
Next, we show some properties of the Wulff shape in the following propositions.
Proposition 4.5 
Suppose that ν is a Borel measure on and that , f are positive continuous functions on . If uniformly on , then .
Proposition 4.6 Suppose that ν is a Borel measure on and that f, g are positive continuous functions on , then .
Then . Therefore, .
Fix , then . By the arbitrariness of , we have for all . Hence . Therefore, . □
Proposition 4.7 Suppose that ν is a Borel measure on and that f, g are positive continuous functions on , then .
Proof Assume that . Fix , then or , thus . By the arbitrariness of , we have for all . Hence . □
Proposition 4.8 Suppose that ν is a Borel measure on and that f, g are positive continuous functions on , then for real number we have .
Hence for all . Therefore, by (2.5), we have . □
Proposition 4.9 Suppose that ν is a Borel measure on and that f, g are positive continuous functions on , then .
Proof Let . Assume , where and , then by (2.5) we have and for all . Thus for all . That is, for all . Therefore, . □
The authors express their deep gratitude to the referees for their many valuable suggestions and comments. The research of QIXIA LIU and GANGSONG LENG was supported by National Natural Science Foundation of China (10971128), and Shanghai Leading Academic Discipline Project (S30104).
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