A characterization of simplices in terms of visibility (Q1300327)

Let \(K\subset {\mathbb R}^d\) be a compact convex set with non-empty interior (convex body). A set \(X\subset {\mathbb R}^d\setminus K\) is said to see the whole boundary of \(K\) if for each boundary point \(z\) of \(K\) there is some \(x\in X\) such that \(K\) has empty intersection with the open segment \((x,z)\). Extending an observation of \textit{V. Soltan} [see Theorem 1 in Stud. Sci. Math. Hung. 28, No. 3-4, 473-483 (1993; Zbl 0817.52013)], the authors prove that for any point \(x_1\in {\mathbb R}^d\setminus K\) there exist \(d\) points \(x_2,\ldots,x_{d+1}\in{\mathbb R}^d\setminus K\) such that \(\{x_1,\ldots,x_{d+1}\}\) sees the whole boundary of \(K\). The main result of the paper states that a convex body \(K\subset {\mathbb R}^d\) is a simplex if and only if for any point \(x_1\in {\mathbb R}^d\setminus K\) there exists a point \(x_2\in {\mathbb R}^d\setminus K\) such that \(\{x_1,x_2\}\) sees the whole boundary of \(K\).