Two optimisation problems for convex bodies (Q2786586)

The authors derive new results in the spirit of the Santaló diagram. More precisely, they investigate two groups of inequalities relating the volume \(V\), the surface area \(S\), and the integral of mean curvature \(I\) of a convex body in Euclidean 3-space with its minimum width and its circumradius, as well as its diameter and inradius, respectively.NEWLINENEWLINE Within this framework it is shown that the spherical symmetric slices (i.e., the part of a ball bounded by two suitable parallel planes equidistant to the center) are the convex bodies maximizing \(V\), \(S\), and \(I\) when the minimum width and the circumradius are given, and that the symmetric 2-cap bodies (i.e., the convex hull of a ball and two suitable outside points symmetric to the center of the ball) are the ones minimizing \(V\), \(S\), and \(I\) for prescribed diameter and inradius.