Addition and subtraction of homothety classes of convex sets
Let \(S_H\) denote the homothety class generated by a convex set \(S\subset \mathbb R^n: S_H = \{a + \lambda S\,| \, a\in \mathbb R^n,\lambda > 0\}\). The author determines 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). Let \(\text{rec}\,S\) denote the recession cone of a closed convex set \(S\). For a pair of line-free closed convex sets \(B\) and \(C\), the following conditions (1)--(3) are equivalent. (1) \(B_H + C_H\) belongs to a unique homothety class generated by a line-free closed convex set. (2) \(B_H + C_H\) lies in the union of countably many homothety classes generated by line-free closed convex sets. (3) There is a line-free closed convex set \(A\) such that: (a) \(\text{rec}\, A = \text{rec}\, B + \text{rec}\, C\), (b) each of the sets \(B_0= B + \text{rec}\, A\) and \(C_0 =C + \text{rec}\, A\) is homothetic either to \(A\) or to \(\text{rec}\, A\), (c) if \(A\) is not a cone, then at least one of the sets \(B_0\), \(C_0\) is not a cone. For a pair of compact convex sets \(B\) and \(C\), each of the conditions (1)--(3) holds iff \(B\) and \(C\) are homothetic. Given \(n\)-dimensional closed convex sets \(B\) and \(C\), we put \[ B_H{\underset {n}\sim} C_H =\{B' \sim C'\, | \, B'\in B_H,\, C'\in C_H,\, \dim (B' \sim C') = n\}. \] Remind the notion of tangential set introduced by R.~Schneider: a closed convex set \(D\) of dimension \(n\) is a tangential set of a convex body \(F\) provided \(F\subset D\) and through each boundary point of \(D\) there is a support hyperplane to \(D\) that also supports \(F\). For a pair of convex bodies \(B\) and \(C\), the following conditions (1)--(4) are equivalent: (1) \(B_H{\underset {n}\sim} C_H\subset B_H\), (2) \(B_H {\underset {n}\sim} C_H\) lies in a unique homothety class generated by a convex body, (3) \(B_H {\underset {n}\sim} C_H\) lies in the union of countably many homothety classes generated by convex bodies, (4) \(B\) is homothetic to a tangential set of \(C\).
scientific article; zbMATH DE number 5117808