ℝ3

Hu Yan

doi:10.1186/1029-242x-2012-3

Abstract

A special case of Mahler volume for the class of symmetric convex bodies in ℝ 3 is treated here. It is shown that a cube has the minimal Mahler volume and a cylinder has the maximal Mahler volume for all generalized cylinders. Further, the Mahler volume of bodies of revolution obtained by rotating the unit disk in ℝ 2 is presented. 2000 Mathematics Subject Classification: 52A20; 52A40.

Highlights

1 Introduction Throughout this article a convex body K in Euclidean n-space Rn is a compact convex set that contains the origin in its interior
Is called the volume product of K, where V (K) denotes n-dimensional volume of K, which is known as the Mahler volume of K, and it is invariant under linear transformation
One of the main questions still open in convex geometric analysis is the problem of finding a sharp lower estimate for the Mahler volume of a convex body K

Summary

Introduction

Throughout this article a convex body K in Euclidean n-space Rn is a compact convex set that contains the origin in its interior. Nazaeov et al [10] proved that the cube is a strict local minimizer for the Mahler volume in the class of origin-symmetric convex bodies endowed with the Banach-Mazur distance. Definition 1 In three-dimensional Cartesian coordinate system OXYZ, if C′ is an origin-symmetric convex body in coordinate plane YOZ, the set:. Where C′ is an origin-symmetric convex body in coordinate plane YOZ, C* is a generalized bicone with vertices (-1, 0, 0) and (1, 0, 0) and the base (C′)*. Because that (C|u⊥)* is a diamond with vertices (-1, 0, 0) and (1, 0, 0), C* | u⊥ is a diamond with vertices (-1, 0, 0) and (1, 0, 0) for any u Î S1, which implies that C* is a bicone with vertices (-1, 0, 0) and (1, 0, 0)

In view of

We can easily prove