Lower bounds for the numbers of antipodal pairs and strictly antipodal pairs of vertices in a convex polytope
Let \(X\) be a finite set of points in a \(d\)-dimensional space \(E^ d\). \textit{V. Klee} [`Unsolved problems in intuitive geometry,' Mimeographed notes, Seattle (1960)] calls points \(x\) and \(y\) from \(X\) antipodal if there are two distinct parallel hyperplanes \(H_ x\) and \(H_ y\) supporting the polytope \(P = \text{conv} X\) such that \(x\) and \(y\) belong to \(H_ x\) and \(H_ y\), respectively. If, additionally, \(P \cap H_ x = \{x\}\) and \(P \cap H_ y = \{y\}\) then the point pair \(x,y\) is called strictly antipodal [cf. \textit{B. Grünbaum}, Isr. J. Math. 1, 5-10 (1963; Zbl 0192.266)]. There are some results on upperbounds of points of a convex body or an \(n\)-pointed set, every pair of which is antipodal. In this paper the greatest lower bounds for the number of antipodal pairs \((r)\) (strictly antipodal pairs \((s))\) of points of a finite set in \(E^ d\) are determined. Avoiding degenerate situations it suffices to consider a convex polytope \(P_ n\) in \(E^ d\) with \(n\) vertices which describe that finite point set. For \(d=2\) the values \(r(P_ n)\) and \(s(P_ n)\) can be given exactly. From these an inequality of B. Grünbaum follows immediately. For \(P_ n\) in \(E^ d\), \(d \geq 3\) the authors give the inequalities \(r(P_ n) \geq n - 1 + d(d - 1)/2\) and \(s(P_{2k}) \geq k\), \(s(P_{2k + 1}) \geq k + 1\) and also all the cases for equalities. To prove this they consider families of convex cones with common zero apex and translate their problem of points into that of cones.
scientific article; zbMATH DE number 536872
scientific article