New counterexamples to A. D. Alexandrov’s hypothesis

Abstract

The paper presents a series of principally di¤erent C y -smooth counterexamples to the following hypothesis on a characterization of the sphere: Let K H R 3 be a smooth convex body. If at every point of qK, we have R 1 c C c R 2 for a constant C, then K is a ball. (R 1 and R 2 stand for the principal curvature radii of qK.)The hypothesis was proved by A. D. Alexandrov and H. F. Mü nzner for analytic bodies. For the case of general smoothness it has been an open problem for years. Recently, Y. Martinez-Maure has presented a C 2 -smooth counterexample to the hypothesis.

Around A. D. Alexandrov’s uniqueness theorem for convex polytopes

“…Firstly, the existence of such a surface (even without additional properties) is far from obvious. However, some candidates already exist in the literature, see [7,9]. Secondly, the existence of a regular triangulation is a strong condition: numerical experiments showed that all previously constructed saddle surfaces either cannot be regularly triangulated, or the existence of a regular triangulation is hard to establish.…”

“…The ingredients of the proofs are the theory of virtual polytopes [14,9,10], the combinatorial rigidity theory, and the theory of pseudo triangulations [15,5] applied to the spherical case in [13]. In the proofs we essentially use three different representations of virtual polytopes: as convex chains, as piecewise linear (support) functions, and as stressed spherically embedded graphs.…”

“…The first counterexample was presented by Y. Martinez-Maure [8]. Later, the author of the paper presented a series of new counterexamples based on the theory oí hyperbolic virttml polytopes (see Appendix A and [9]). …”

Pseudo-triangulations—a survey

“…A different (and simpler) proof of 7.7 is given in Section 7.4 for the case of Laman graphs. 44 GÜNTER ROTE, FRANCISCO SANTOS, AND ILEANA STREINU 7.2. Stretching Combinatorial Pseudo-Triangulations.…”

“…in the area of art galleries (illumination by floodlights) [61]; and in an area which, at first sight, may seem unrelated to discrete geometry: the construction of counter-examples to a conjecture of A. D. Alexandrov characterizing the sphere among all smooth surfaces [44].…”