The Convex Hull of Several Compacta

Abstract

Let K₀,K₁,...,K_m be nonempty compact sets in $\mathbb{R}^n$. Then the convex hulls ${\bigcup_{i=0}^m (K_i + r_i)}$ with r₀=0 form a convex family of sets parametrized by $\rho = (r_1,\ldots,r_m) \in \mathbb{R}^nm$. For m=1, the volume Volconv(K₀∪(K₁+r)) is a convex function of $r \in \mathbb{R}^n$. Bibliography: 5 titles.