Separation in semi-cyclic 4-polytopes
Let \(F_1\) and \(F_2\) be two faces of a \(4\)-dimensional simplicial polytope \(P\) with \(F_1\subset F_2\) and \(\dim(F_1)=1\), \(\dim(F_2)=3\). Let \({\mathcal F}\) be the set of all facets \(F\) of \(P\) with \(F_1\subset F\not= F_2\). Then there is a point \(x\) that lies beyond all facets in \({\mathcal F}\) and beneath all other facets. The polytope \(P^+:=\text{conv}(P\cup\{x\})\) is said to be obtained from \(P\) by sewing \(x\) through \({\mathcal F}\). This construction is due to \textit{I. Shemer} [Isr. J. Math. 43, 291-314 (1982; Zbl 0598.05010)]. Semi-cyclic polytopes are the ones that are obtained via iterated sewing operations, starting from any simplex \(\text{conv}\{x_1,\dots,x_5\}\), and successively adding the vertices \(x_6,\dots,x_n\), where at the iteration adding \(x_n\) the edge \(F_1=\text{conv}\{x_1,x_{n-1}\}\) is used, and \(F_2\supset F_1\) is chosen arbitrarily. If \(F_2=\text{conv}\{x_1,x_2,x_{n-2},x_{n-1}\}\) is chosen at each iteration, then one obtains cyclic \(4\)-polytopes with \(n\) vertices. The main theorem of the paper states that for each semi-cyclic \(4\)-polytope \(P\subset{\mathbb R}^4\), and for each interior point \(x\) of \(P\), there are nine affine hyperplanes in \({\mathbb R}^4\) such that every facet of \(P\) is separated from \(x\) by one of them. This was previously known for cyclic \(4\)-polytopes due to \textit{K. Bezdek} and \textit{T. Bisztriczky} [Geom. Dedicata 68, No. 1, 29-41 (1997; Zbl 0965.52014)]. The result implies that the polars of semi-cyclic \(4\)-polytopes can be covered by at most nine smaller positively homothetic copies. In particular, they satisfy the Gohberg-Markus-Hadwiger covering conjecture.
scientific article; zbMATH DE number 1809798