2023/06/20 by René Brandenberg, Brandenberg, René, Katherina von Dichter +3 · 1 citation
Mathematics · #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2306.11460
For a Minkowski centered convex compact set K we define α(K) to be the smallest possible factor to cover K ∩ (-K) by a rescalation of conv (K∪ (-K)) and give a complete description of the possible values of α(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].