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

Computational Aspects of the Hausdorff Distance in Unbounded Dimension

We study the computational complexity of determining the Hausdorff distance of two polytopes given in halfspace- or vertex-presentation in arbitrary dimension. Subsequently, a matching problem is investigated where a convex body is allowed to be homothetically transformed in order to minimize its Hausdorff distance to another one. For this problem, we characterize optimal solutions, deduce a Helly-type theorem and give polynomial time (approximation) algorithms for polytopes

Successive Radii and Ball Operators in Generalized Minkowski Spaces

We investigate elementary properties of successive radii in generalized Minkowski spaces (that is, with respect to gauges), i.e., we measure the "size" of a given convex set in a finite-dimensional real vector space with respect to another convex set. This is done via formulating some kind of minimal containment problems, where intersections or Minkowski sums of the latter set and affine flats of a certain dimension are incorporated. Since this is strongly related to minimax location problems and to the notions of diametrical completeness and constant width, we also have a look at ball intersections and ball hulls.Comment: submitted to "Advances of Geometry