Minimization of the inradius of convex bodies for prescribed diameter and circumradius in Minkowski spaces

Abstract

We study Blaschke-Santaló diagrams for the inradius, circumradius, and diameter in arbitrary-dimensional Minkowski spaces. Their boundary parts are regularly described by four linear inequalities and one more complex inequality, and we analyze which bodies fill the boundaries and for which (types of) norms that boundaries collapse to a single point. In this context, the diagram with respect to the 1-norm in 3-space plays an exceptional role. We complete the description of the according diagram by tightening Bohnenblust's inequality through involvement of the inradius.