A problem of McMullen on the projective equivalences of polytopes
In terms of convexity for oriented matroids the authors give a version of the following problem of McMullen [cf. Contributions to geometry, Proc. Symp., Siegen 1978, 92-130 (1979; Zbl 0445.52006)]: ''Determine the largest number \(\partial (d)\) of points, lying in general position in \(R^ d\), which may be mapped by a permissible projective transformation onto the set of vertices of a convex polytope''. It is proved, that for every orientation \(\Theta\) of the uniform matroid \(U_{2r-1,r}\) with \(r\geq 3\), there is an acyclic reorientation, such that all 2r-1 points are extreme. This result generalizes the inequality of Larman \(\partial (r-1)\geq 2r-1\) for the above problem of McMullen.
scientific article; zbMATH DE number 3915110