Farthest points and cut loci on some degenerate convex surfaces (Q1882449)

The authors investigate shortest paths (with respect to Euclidean distance) and cut loci for two special classes of abstract polytopal complexes: (1) for the (degenerate) cell decompositions of \(S^n\), obtained by gluing together two isometric \(n\)-simplices along their common boundary; (2) for the (degenerate) cell decompositions of \(S^2\), obtained by gluing together two copies of a convex polygon. In the first case, the cut loci turn out to be unions of lower dimensional simplices that either are all faces of the complex or have a common point. In the second case, the cut loci are segment trees (i.e., trees that are unions of line segments). In particular, every combinatorial type of a tree can be realized as the cut locus of some point on some union of two isometric convex polygons.