268 research outputs found
Sharpening Geometric Inequalities using Computable Symmetry Measures
Many classical geometric inequalities on functionals of convex bodies depend
on the dimension of the ambient space. We show that this dimension dependence
may often be replaced (totally or partially) by different symmetry measures of
the convex body. Since these coefficients are bounded by the dimension but
possibly smaller, our inequalities sharpen the original ones. Since they can
often be computed efficiently, the improved bounds may also be used to obtain
better bounds in approximation algorithms.Comment: This is a preprint. The proper publication in final form is available
at journals.cambridge.org, DOI 10.1112/S002557931400029
An overview of complex ellipsoids
An ellipsoid is the image of a ball under an affine transformation. If this
affine transformation is over the complex numbers, we refer to it as a complex
ellipsoid. Characterizations of real ellipsoids have received much attention
over the years however, characterizations of complex ellipsoids have been
studied very little. This paper is a review of what is known about complex
ellipsoids from the point of view of convex geometry. In particular, the proof
of the Complex Banach Conjecture.Comment: arXiv admin note: substantial text overlap with arXiv:2111.1428
On the volume of the convex hull of two convex bodies
In this note we examine the volume of the convex hull of two congruent copies
of a convex body in Euclidean -space, under some subsets of the isometry
group of the space. We prove inequalities for this volume if the two bodies are
translates, or reflected copies of each other about a common point or a
hyperplane containing it. In particular, we give a proof of a related
conjecture of Rogers and Shephard.Comment: 9 pages, 3 figure
- …