Upper bounds for edge-anitpodal and subequilateral polytopes (Q2466338)

A convex \(d\)-polytope \(P\) is called edge-antipodal if for each edge \(xy\) of \(P\), vertices \(x\) and \(y\) lie on distinct parallel supporting planes of \(P\). The author defines a polytope \(P\) to be subequilateral if the length of each of its edges equals its diameter. The author's main result is the explicit upper bound \((d/2 +1)^d\) for the number of vertices of an edge-antipodal or a subequilateral \(d\)-polytope \((d \geq 2)\). The bound is not sharp for \(d \geq 3\), see \textit{K. Bezdek}, \textit{T. Bisztriczky} and \textit{K. Böröczky} [Combinatorial and computational geometry, Cambridge University Press, pp. 129--134 (2005; Zbl 1091.52005)]. The author also shows that a subequilateral \(d\)-polytope in Euclidean, hyperbolic or elliptic \(d\)-space is an equilateral simplex.