Das isodiametrische Problem der Minkowski-Geometrie. (The isodiametric problem in Minkowskian geometry) (Q1096184)

The paper deals with some inequalities in the style of the classical inequalities of Minkowski for convex bodies: a) If A and B are two non-void compact sets in affine space \(A^ n\) with standard body (Eichfigur) (o,E) the inequality \(| A|^{1/n}+ | B|^{1/n}\leq D(A,B;E)| E|^{1/n}\) holds, where \(| X|\) means the Lebesgue measure of X and D(A,B;E) is the maximal distance between A and B with the Minkowski metric induced by the standard body (o,E); b) For any non-void measurable set A in affine space \(A^ n\) with standard body (o,E), the inequality \(D(A,E)\geq 2| E|^{1/n}| A|^{1/n}\) holds. The cases of inequality are discussed and several consequences are pointed out. For \(n=3\), noteworthy applications to computer images related with crystallography are given.