A case when the union of polytopes is convex (Q1774983)

The authors present the following necessary and sufficient condition for the union of a finite number of convex polytopes in \(\mathbb{R}^d\) to be convex. Let \(P_1,\ldots,P_n\) (\(n\geq 2\)) be convex polytopes with vertex-sets \(X_1,\ldots,X_n\), respectively, and let \(P:=\bigcup_{i=1}^{n}P_i\), \(X:=\bigcup_{i=1}^{n}X_i\) and \(Q:=\text{conv}(X)\) (convex hull of \(X\)). Then \(P=Q\) if and only if \(\text{conv}(S) \subset P\) for each \(S\subset X\) with \(| S| \leq d+1\) and \(| S\cap X_i| \leq 1\) for \(i=1,\dots,n\).