A note on the minimum number of edge-directions of a convex polytope
The minimum number of different edge-directions of a convex \(n\)-gon is obviously \(\lceil\frac n2\rceil\). For higher dimensions, the authors prove that for any convex \(d\)-polytope with \(n\) vertices the number of distinct edge-directions is bounded from below by \(\frac{d-1}{2e} n^{1/ (d-1)}\), this lower bound being asymptotically tight and attained, up to a multiplicative factor, e.g., by zonotopes generated by points on the moment curve, and also by any cubical \(d\)-zonotope. From this the following corollary is obtained (regardless of the dimension of the polytope): The number of different edge-directions of any convex polytope with \(n\) vertices is \(\tfrac 12\log_en\).
scientific article; zbMATH DE number 2083265