Moscow--Petersburg seminar on low-dimensional mathematics
September 10, 2004

On A.D.Alexandrov's hypothesis, hyperbolic polytopes and hyperbolic fans
G.Panina

Non-trivial hyperbolic virtual polytopes appeared as an auxillary
construction for various counterexamples to the following A.D.Alexandrov's
conjecture:

Let  $K \subset \Bbb R^3$ be a smooth convex body If for a constant $C$,
in each point of $ \partial K$, we have $R_1 \leq C \leq R_2$, then $K$ is 
a ball. ($R_1$ and $R_2$ are the principal curvature radii of $\partial K$).}

For a long time mathematicians were certain about correctness of the
hypothesis but obtained only some partial resunts.  Recently,
Y.Martinez-Maure (2001) has given a counterexample. First, he demonstrated
that each smooth {\it hyperbolic h\'erisson } (i.e., a hyperbolic h\'erisson
with a smooth support function) generates a desired counterexample. Next, he
presented such an example, namely, a hyperbolic surface with four {\it horns} 
(i.e., points where the surface is neither hyperbolic nor smooth), given by
an explicit formula.

Surprisingly, this counterexample proved to be not unique: a series of
counterexamples was given by G.Panina (2003) (hyperbolic h\'erissons with
any even number of horns).

Later, it turned out that they are even more various. advanced examples (in
particular, with odd number of horns) were obtained by G.Panina, 2004.

In addition, we discuss the fans of hyperbolic virtual polytopes.
They have interesting combinatorial properties.

The edges of such a fan admit a {\it proper coloring}, which encodes
important properties of the virtual polytope. For example, a cell of the fan
corresponds to a horn of the polytope, if and only if the color changes two
times as going around the cell.

Hyperbolic virtual polytopes can be classified in a reasonable way by the
number of horns. However, there exists a finer classification related to
arrangements of non-intersecting oriented great semi-circles on the sphere.

Regular triangulations of hyperbolic fans are of particular interest. They
lead to a refinement of A.D. Alexandrov's uniqueness theorem for convex
polytopes.

---
http://www.pdmi.ras.ru/~lowdimma