Sets which can be extended to 𝑚-convex sets

Abstract

Let S be a compact set in R d , T 0 ⊆ S {R^d},{T_0} \subseteq S . Then T 0 {T_0} lies in an m-convex subset of S if and only if every finite subset of T 0 {T_0} lies in an m-convex subset of S. For S a closed set in R d {R^d} and T 0 ⊆ S {T_0} \subseteq S , let T 1 = { P : P {T_1} = \{ P:P a polytope in S having vertex set in T 0 , dim ⁡ P ⩽ d − 1 } {T_0},\dim P \leqslant d - 1\} . If for every three members of T 1 {T_1} , at least one of the corresponding convex hulls \[ conv { P i ∪ P j } , 1 ⩽ i > j ⩽ 3. {\text {conv}}\{ {P_i} \cup {P_j}\} ,\quad 1 \leqslant i > j \leqslant 3. \] lies in S, then T 0 {T_0} lies in a 3-convex subset of S. An analogous result holds for m-convex sets provided ker S ≠ ∅ S \ne \emptyset .