Sets which can be extended to $m$-convex sets

Abstract:

Let S be a compact set in ${R^d},{T_0} \subseteq S$. Then ${T_0}$ lies in an m-convex subset of S if and only if every finite subset of ${T_0}$ lies in an m-convex subset of S. For S a closed set in ${R^d}$ and ${T_0} \subseteq S$, let ${T_1} = \{ P:P$ a polytope in S having vertex set in ${T_0},\dim P \leqslant d - 1\}$. If for every three members of ${T_1}$, at least one of the corresponding convex hulls \[ {\text {conv}}\{ {P_i} \cup {P_j}\} ,\quad 1 \leqslant i < j \leqslant 3.\] lies in S, then ${T_0}$ lies in a 3-convex subset of S. An analogous result holds for m-convex sets provided ker $S \ne \emptyset$.