We first see that, in the sense of Baire categories, many convex surfaces have quite large cut loci and infinitely many relative maxima of the distance function from a point. Then we find that, on any convex surface, all these extreme points lie on a single subtree of the cut locus, with at most three endpoints. Finally, we confirm (both in the sense of measure and in the sense of Baire categories) Steinhaus’ conjecture that “almost all" points admit a single farthest point on the surface.

The farthest point mapping on compact surfaces, associating to each point x of the surface the set of absolute maxima of the intrinsic distance from x, is for some surfaces single-valued and a homeomorphism, while for other surfaces it is not single-valued, and not surjective. These two big classes are not very well understood. For instance it is still unknown whether, say in the convex case, the second class is dense. For a C² metric on both surfaces and the space of surfaces, the first class has, however, nonempty interior. We describe various properties of the sets of critical points, and of relative and absolute maxima of distance functions, and find several connections between them. We see for example that, on smooth surfaces homeomorphic to S², a point cannot be critical with respect to more than one other point. Sufficient conditions for a surface to belong to the second class will be formulated and a particular Tannery surface belonging to the boundary of both classes will be presented.

Consider the mapping F associating to each point x of a convex surface the set of all points at maximal intrinsic distance from x. We provide a large class of surfaces on which F is single-valued and involutive. Moreover, we show that there are point-symmetric surfaces of revolution with F single-valued but not involutive.

We say that a polytope has the Rupert property if we can make a hole large enough in to permit another copy of to pass through. In this article, we show that among the 13 Archimedean solids, 8 have this property, namely, the cuboctahedron, the truncated octahedron, the truncated cube, the rhombicuboctahedron, the icosidodecahedron, the truncated cuboctahedron, the truncated icosahedron, and the truncated dodecahedron.