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

Extremal solutions for Network Flow with Differential Constraints -- A Generalization of Spanning Trees

In network flow problems, there is a well-known one-to-one relationship between extreme points of the feasibility region and trees in the associated undirected graph. The same is true for the dual differential problem. In this paper, we study problems where the constraints of both problems appear simultaneously, a variant which is motivated by an application in the expansion planning of energy networks. We show that all extreme points still directly correspond to graph-theoretical structures in the underlying network. The reverse is generally also true in all but certain exceptional cases. We furthermore characterize graphs in which these exceptional cases never occur and present additional criteria for when those cases do not occur due to parameter values

From inequalities relating symmetrizations of convex bodies to the diameter-width ratio for complete and pseudo-complete convex sets

For a Minkowski centered convex compact set $K$ we define $\alpha(K)$ to be the smallest possible factor to cover $K \cap (-K)$ by a rescalation of $\mathrm{conv} (K\cup (-K))$ and give a complete description of the possible values of $\alpha(K)$ in the planar case in dependence of the Minkowski asymmetry of $K$. As a side product, we show that, if the asymmetry of $K$ is greater than the golden ratio, the boundary of $K$ intersects the boundary of its negative $-K$ always in exactly 6 points. As an application, we derive bounds for the diameter-width-ratio for pseudo-complete and complete sets, again in dependence of the Minkowski asymmetry of the convex bodies, tightening those depending solely on the dimension given in a recent result of Richter [10]

Radii of regular polytopes

For the first time complete lists of two pairs of inner and outer radii classes of the three types of regular polytopes which exist in all dimensions are presented. A new access using isotropic polytopes provides easier understanding of the underlying geometry and helps unifying the results