On the quermassintegrals of convex bodies
© Zhao and Cheung; licensee Springer 2013
Received: 18 March 2013
Accepted: 9 May 2013
Published: 27 May 2013
The well-known question for quermassintegrals is the following: For which values of and every pair of convex bodies K and L, is it true that
In 2003, the inequality was proved if and only if or . Following the problem, in the paper, we prove some interrelated results for the quermassintegrals of a convex body.
In particular, , .
A discussion of the cases of equality is contained in the reference .
There is a remarkable similarity between inequalities about symmetric functions (or determinants of symmetric matrices) and inequalities about the mixed volumes of convex bodies. For example, the analogue of (1.2) in the Brunn-Minkowski theory is as follows.
with equality if and only if K and L are homothetic, where is the i th quermassintegral of K (see Section 2).
In 1991, the special case was stated also in  as an open question. In the same paper it was also mentioned that (1.6) follows directly from the Aleksandrov-Fenchel inequality when and L is a ball.
In 2002, it was proved in  that (1.6) is true for all in the case where L is a ball.
In 2003, it was proved in  that (1.6) holds true for every pair of convex bodies K and L in if and only if or .
is true for every pair of convex bodies K and L in if and only if or .
In this paper, following the above results, we prove the following interest results.
is a convex function on if and only if or .
if and only if or .
2 Notations and preliminaries
The setting for this paper is an n-dimensional Euclidean space . Let denote the set of convex bodies (compact, convex subsets with non-empty interiors) in . We reserve the letter u for unit vectors, and the letter B for the unit ball centered at the origin. The surface of B is . The volume of the unit n-ball is denoted by .
where denotes the usual inner product u and x in .
Let δ denote the Hausdorff metric on , i.e., for , , where denotes the sup-norm on the space of continuous functions .
If is positive and continuous, K will be called a star body. Let denote the set of star bodies in . Let denote the radial Hausdorff metric, as follows, if , then .
where the sum is taken over all n-tuples () of positive integers not exceeding r. The coefficient depends only on the bodies and is uniquely determined by (2.1). It is called the mixed volume of , and is written as . Let and , then the mixed volume is written as . If , , then the mixed volume is written as and is called the quermassintegral of a convex body K.
known as formula ‘Steiner decomposition’.
3 Proof of main results
Hence the function is a convex function on for every star body K and L if and only if or . □
if and only if or .
if and only if or .
Letting , we conclude Theorem 1.2. □
First author is supported by the National Natural Science Foundation of China (10971205). Second author is partially supported by a HKU URG grant.
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