Institute of
Discrete Mathematics
 and Geometry

Workshop 2:
Herissons and Virtual Polytopes

Roughly speaking, a virtual polytope is the Minkowski difference of two convex compact polytopes, and a herisson (or a hedgehog) is its geometrical version adopted for the Euclidean 3-space. The idea to use the Minkowski difference of convex bodies may be traced back to some papers by A.D. Alexandrov and H. Geppert in the 1930's. This notion was made precise in different ways in the 1990's. It arose from some problems of algebraic geometry, of the theory of convex bodies, the theory of singularities, the theory of minimal surfaces, and of Cauchy-type rigidity theorems for polyhedra, etc.

A related notion for smooth objects enabled Y. Martinez-Maure to prove that the following theorem by A.D. Alexandrov known since 1937 for analytical surfaces is wrong for C-surfaces: "Every analytical convex surface in the Euclidean 3-space whose principal curvatures k1 and k2 satisfy the inequality (k1 - c)(k2 - c) ≤ 0 with some constant c must be a sphere".

The aim of this workshop is to present recent developments in virtual polytopes and herissons.

Last update March 30, 2005