Antipodes sur un t�tra�dre r�gulier

Abstract

The aim of this article is to compute the set of farthest points of an arbitrary point of the surface of a regular tetrahedron endowed with its intrinsic metric.2000 Mathematics Subject Classification: 52B10, 51N05, 51N20, 51M04, 52B55.

Farthest points and cut loci on some degenerate convex surfaces

“…H. Steinhaus asked explicitly for characterizations of these sets on convex surfaces (see the chapter A35 of [3]); recently, T. Zamfirescu ([14], [15], [16], [17]) established several properties of the local and global maxima of distance functions on convex surfaces, thus answering H. Steinhaus' questions. Some of his results were improved by J. Rouyer [10], under the stronger assumption of convex polyhedral surfaces.…”

On antipodes on a convex polyhedron

“…C. Vîlcu disproved this conjecture in [4]. However, the regular tetrahedron does not provide a counterexample (refer to [2]). It is not clear yet whether there exists any polyhedral counterexample.…”

“…In the special case of a regular tetrahedron, we can compute the equations of the zone lines explicitly [2].…”

Criteria for farthest points on convex surfaces

“…J. Rouyer [15] determined explicitly the set F for (the boundary of ) a regular tetrahedron T ; in particular, each vertex v of T has curvature π, belongs to F, F −1 v is a tree with three leaves, and F is arcwise connected. Example 2.5 Theorem 2.1 directly applies to tetrahedra.…”

“…Despite the simplicity of the notion, few examples of completely determined sets of farthest points seem to be known (see [8], [9], [15], [17], [20]), a possible reason being the difficulty to determine by direct computation these sets on general surfaces. This "quasi-poverty" constitutes a first motivation for presenting criteria to recognize farthest points, or caps to which they belong, on convex surfaces.…”