Minimal and reduced pairs of convex bodies
Call two pairs of convex bodies \((K,M)\) and \((K',M')\) in \(\mathbb{R}^d\) equivalent if the Minkowski sums \(K'+M\) and \(K+M'\) are equal. A pair \((K,M)\) is minimal if there is no equivalent pair \((K',M')\) with \(K'\varsubsetneq K\), \(M'\subseteq M\) or \(K'\subseteq K\), \(M'\varsubsetneq M\). A pair \((K,M)\) is called reduced if, whenever \(K'+M=K+M'=C\), there is a body \(L\) such that \(K+M+L=C\); reduced pairs are minimal. The author gives a new proof, using surface area measures, (and avoiding the use of Zorn's lemma) that equivalence classes of pairs of convex bodies in the plane always contain minimal elements, and a new proof of uniqueness of minimal pairs (up to translation). New characterizations of minimal and reduced planar pairs are also given, in terms of surface area measures. As a corollary of the first of these, it is shown that a pair of convex polygons is minimal iff then have at most one pair of equiparallel edges. In the final section, this corollary is generalized to pairs of polytopes in higher-dimensional spaces. Conditions for minimality are given, and it is shown that the reduced pairs of polytopes are precisely those with no equiparallel edges. Using the density of polytopes, it is shown that most pairs of convex bodies in \(\mathbb{R}^d\) are reduced, and the unique (up to translation) minimal members of their equivalence class.
scientific article; zbMATH DE number 929898