On the convex hull of several compacta

A. V. Evdokimov ; V. A. Zalgaller

Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 4, Tome 261 (1999), pp. 66-75

Résumé

Let $K_0,K_1,\dots,K_m$ be nonempty compact sets in $\mathbb R^n$. Then the family of convex hulls $\operatorname{conv}\{\bigcup^m_{i=0}(K_i+r_i)\}$, $r_0=0$, is a convex family of sets, parametrized by $\rho=(r_1,\dots,r_m)\in\mathbb R^{nm}$. In case $m=1$, the volume $\operatorname{Vol\,conv}(K_0\cup(K_1+r))$ is a convex function of $r\in\mathbb R^n$.