On typical degenerate convex surfaces

Abstract

Various properties are given concerning geodesics on, and distance functions from points in, typical degenerate convex surfaces; i.e., surfaces obtained by gluing together two isometric copies of typical (in the sense of Baire category) convex bodies, by identifying the corresponding points of their boundaries. Mathematics Subject Classification (2000) 52A20 · 53C45

What do cylinders look like?

“…The limiting case h = 0 in Theorem 3.1 provides a differentdirect-proof for the last part of Theorem 1 in [13]. Theorem 3.1 can also be used (see §3 in [13]) to show the existence of convex surfaces S ⊂ R 3 (e.g., cylinders with 3 planar symmetries), such that S has a closed curve O most points of which are endpoints of S, and also has a simple closed geodesic crossing O. Other examples of such cut loci have been obtained in [3,6,10,13,15,17].…”