Addition and subtraction of homothety classes of convex sets

Valeriu Soltan

Abstract: Let $S_H$ denote the homothety class generated by a convex set $S \subset {\mathbb R}^n$: $S_H = \{a + \lambda S \mid a \in {\mathbb R}^n, \lambda > 0\}$. We determine conditions for the Minkowski sum $B_H + C_H$ or the Minkowski difference $B_H \sim C_H$ of homothety classes $B_H$ and $C_H$ generated by closed convex sets $B,C \subset {\mathbb R}^n$ to lie in a homothety class generated by a closed convex set (more generally, in the union of countably many homothety classes generated by closed convex sets).

Keywords: convex set, homothety class, Minkowski sum, Minkowski difference

Classification (MSC2000): 52A20

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