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 3space.
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 Cauchytype rigidity theorems for polyhedra, etc.
A related notion for smooth objects enabled Y. MartinezMaure 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 3space whose principal
curvatures k_{1} and k_{2} satisfy the inequality
(k_{1}  c)(k_{2}  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
