\(L_{p}\) Harmonic radial combinations of star bodies
- Rulin Shen^{1} and
- Baocheng Zhu^{1}Email author
https://doi.org/10.1186/s13660-015-0654-z
© Shen and Zhu; licensee Springer. 2015
Received: 5 October 2014
Accepted: 1 April 2015
Published: 16 April 2015
Abstract
For star bodies, the \(L_{p}\) harmonic radial combinations were defined and studied in several papers. In this paper, we study the mean chord of \(L_{p}\) harmonic radial combinations of star bodies and get an upper bound for dual mixed volumes of \(L_{p}\) harmonic radial combination of star bodies and their polar bodies. Furthermore, we obtain a dual Urysohn type inequality and a dual Bieberbach type inequality.
Keywords
MSC
1 Introduction
The classical Brunn-Minkowski theory originated with Minkowski when he combined his concept of mixed volume with the Brunn-Minkowski inequality, which is the core of convex geometric analysis. This theory was developed from a few basic concepts such as support function, vector addition, and volume. Since Firey introduced his new \(L_{p}\) addition in 1960s (see [1]), the new \(L_{p}\) Brunn-Minkowski theory was born in Lutwak’s papers [2, 3] and it has witnessed a rapid growth (see, e.g., [4–13]).
In the 1970s, Lutwak introduced the dual mixed volume and hence developed the dual Brunn-Minkowski theory, which helped achieving a major breakthrough in the solution of the Busemann-Petty problem in the 1990s. The \(L_{p}\) harmonic radial combination of convex bodies was first investigated by Firey (see [14, 15]). Then, the \(L_{p}\) harmonic radial combination was extended to star bodies by Lutwak [3], and it plays a key role in the dual \(L_{p}\) Brunn-Minkowski theory.
For star bodies, the \(L_{p}\) harmonic radial combination was introduced and studied in several papers (see, e.g., [1, 3, 16–20]). The aim of this paper is to study them further, that is, we mainly investigate the mean chord of \(L_{p}\) harmonic radial combination of star bodies.
Let ${\mathcal{K}}^{n}$ denote the set of convex bodies (compact, convex subsets with nonempty interiors) in \(\mathbb{R}^{n}\) and ${\mathcal{K}}_{o}^{n}$ denote the subset of ${\mathcal{K}}^{n}$ consisting of all convex bodies that contain the origin in their interiors. Let ${\mathcal{S}}_{o}^{n}$ denote the set of star bodies (star-shaped, continuous radial functions) in \(\mathbb{R}^{n}\) containing the origin in their interiors. The unit ball in \(\mathbb{R}^{n}\) and its surface will be denoted by B and \(S^{n-1}\), respectively. The volume of B will be denoted by \(\omega_{n}\), the \((n-1)\)-dimensional volume \(\alpha_{n-1}\) of \(S^{n-1}\) is \(\alpha_{n-1}=n\omega_{n}\).
In [22], the mean width of the Firey linear combinations of convex bodies was studied, and the lower bound of the mean width of the Firey linear combinations of convex body and its polar body was given.
In this paper, we give some good properties of \(L_{p}\) harmonic radial combination of star bodies from the definition directly. Besides these properties, we also establish an upper bound for dual mixed volumes \(\widetilde{V}_{i}(\cdot, \cdot)\) of \(L_{p}\) harmonic radial combination of star bodies and their polar bodies as follows.
Theorem 1.1
In [23], Hadwiger defined the mean width \(\overline{b}(K)\) of $K\in {\mathcal{K}}_{o}^{n}$. Here we prove the following.
Theorem 1.2
(Dual Urysohn type inequality)
This immediately yields the following inequality.
Theorem 1.3
(Dual Bieberbach type inequality)
2 Preliminaries
2.1 Mixed volumes and mean width
2.2 Dual mixed volumes and mean chord
3 Main results and proofs
In the following, we obtain some good properties and inequalities for the \(L_{p}\) harmonic radial combinations of star bodies from the definitions directly.
Theorem 3.1
Proof
Theorem 3.2
(Positive multisublinear)
Proof
Just like Theorem 3.2, we have one more general property than that of Theorem 3.1 as follows. It is also the dual of inequality (3.2).
Theorem 3.3
(Positive multisublinear)
Proof
Next, we give the proof of Theorem 1.1 which was illustrated in Section 1. We shall prove a generalized form of an upper bound for the dual mixed volume.
Theorem 3.4
Proof
Remark 3.1
Theorem 1.1 is just the case \(i=\frac{n}{2}\) of Theorem 3.4, and we complete the proof of Theorem 1.1.
In the following, we will obtain a dual Urysohn type inequality and a dual Bieberbach type inequality.
Lemma 3.1
(see [26])
Theorem 3.5
Proof
If we let \(i=0\) in Theorem 3.5, then we have the following.
Corollary 3.1
At the same time, by the last equation in (2.1) we can obtain the dual Urysohn type inequality (see [26] for the dual Urysohn inequality):
Corollary 3.2
(Theorem 1.2)
This immediately yields the dual Bieberbach type inequality (see [26] for the dual Bieberbach inequality):
Corollary 3.3
(Theorem 1.3)
Declarations
Acknowledgements
Authors would like to thank two anonymous referees for many helpful comments and suggestions that directly led to the improvement of the original manuscript. This work is supported in part by the Doctor Starting Foundation of Hubei University for Nationalities (No. MY2014B001) and National Natural Science Foundation of China (No.11201133).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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