Extremal convex planar sets
The authors obtain several inequalities relating the area, the perimeter, the diameter and the minimal width of a planar convex set; they also determine the extremal sets. They try to build a complete system of inequalities using also some classical inequalities proved by Lebesgue, Kubota Fukasawa, Sholander and Hemmi. A system of inequalities concerning \(n\) geometric magnitudes of a set is said to be complete if for any collection of \(n\) numbers satisfiying those conditions, a set with these values of the characteristics exists. This notion was introduced by Blaschke. From a complete system of inequalities all inequalities concerning these magnitudes can be found. Since Blaschke several complete systems concerning three parameters have been obtained (although Blaschke's original problem is still open). This paper represents the first attempt to build a complete system of inequalities concerning four parameters. They provide an alternative version of Blaschke's diagram as the cross-section of a cone; this diagram turns out to be a compact set in 3-dimensional euclidean space whose boundary is a finite family of surfaces each one corresponding to one inequality. The authors succeed in determining nearly all the surfaces; only the bottom boundary is missing, and they present an interesting conjecture for the bottom boundary.
scientific article; zbMATH DE number 2162289