Separation in totally-sewn 4-polytopes with the decreasing universal edge property (Q664204)

The separation number \(s(O,P)\) of a \(d\)-polytope \(P\) with respect to a point \(O\) in the interior of \(P\) is the minimum number of hyperplanes necessary to strictly separate \(O\) from any facet of \(P\). According to a variation of the Gohberg-Markus-Hadwiger conjecture that is known as the Bezdek's separation conjecture, \(s(O,P)\leq 2^d\) for all \(d\)-polytopes. In general this is open but is proved for cyclic \(d\)-polytopes, see [\textit{K. Bezdek} and \textit{T. Bisztriczky}, Geom. Dedicata 68, No. 1, 29--41 (1997; Zbl 0965.52014)]. The authors verify the conjecture for totally-sewn 4-polytopes with the property that the number of universal edges does not increase as the polytopes are constructed. Here a polytope is called totally-sewn if it can be constructed by iterations of the following procedure that starts with a 4-simplex: the next neighbourly 4-polytope is the convex hull of the previous polytope \(P\) and a suitably located exterior point of \(P\).