On convex polytopes in \(\mathbb R^{d}\) containing and avoiding zero (Q1940364)

The authors prove certain inequalities between the numbers of convex polytopes in \(\mathbb{R}^d\) ``containing'' and ``avoiding'' zero provided that their vertex sets are subsets of a given finite set \(S\subset\mathbb{R}^d\). A set \(X\subset\mathbb{R}^d\) is called \(z\)-containing (\(z\)-avoiding) if \(z\in \mathbb{R}^d\) is from \(\text{int\,conv\,}X\) \((z\not\in\text{int\,conv\,}X)\). Let \(C(S)\) and \(A(S)\) denote the sets of minimal \(z\)-containing and maximal \(z\)-avoiding subsets of \(S\), respectively (defined via proper subsets). One of the main results reads as follows: Let \(S\subset\mathbb{R}^d\) be a finite set and \(z\in\mathbb{R}^d\setminus S\). Suppose that \(z\) is in general position with respect to \(S\). Then \(|A(S)|\leq d|C(S)|+ 1\).