A counterexample to A.D. Alexandrov's conjecture EG-Models Home

image KP1.jpg
Figure 1: The graph G
image KP2.gif
Figure 2: Two local colorings
image KP3.gif
Figure 3
image KP4.gif
Figure 4: Hyperbolic virtual polytope
Electronic Geometry Model No. 2010.02.002


Marina Knyazeva and Gaiane Panina


A pointed graph on the sphere which leads to a counterexample to A.D. Alexandrov's conjecture.

We present a graph G embedded in the sphere S2 such that:

It is pretty difficult to find such an example (try by yourself!), but still there are many of them.

This graph is interesting and important not only because of its funny combinatorics, but also because it leads to a counterexample to A.D. Alexandrov's uniqueness conjecture for smooth convex surfaces:

Let K in R3 be a smooth closed convex surface. If a constant C separates non-strictly principal curvature radii at every point of the boundary of K, then K is a ball.

This was proven by A.D. Alexandrov in [1] for analytic surfaces; however, the general case ( for smooth surfaces) stood open for a long time. The first counterexample by Y. Martinez-Maure appeared in 2001.

We explain below how the graph G leads to a counterexample to the conjecture, referring the reader to [2-5] for details.

This example illustrates a small part of the theory of hyperbolic virtual polytopes .

The below brief explanation contains no precise definitions and constructions, for which we refer the reader to [2,4,5]. Virtual polytopes are Minkowski differences of convex polytopes. They can be introduced as just formal expressions, but the key point is that the formal differences can be interpreted geometrically. Basic geometric notions for convex polytopes (such as volume, faces, outer normal fan, etc.) can be extended to virtual polytopes. In particular, it makes sense to speak of the support function of a virtual polytope.

When passing from convex polytopes to virtual polytopes, we loose convexity. Generically, a virtual polytope has a mixed type of convexity: it has a convex part (whatever this means), a concave part, and a saddle part. Virtual polytopes whose support function is saddle are called hyperbolic.

From this viewpoint, the above example presents the support function h of some hyperbolic virtual polytope K. The geometric interpretation of the virtual polytope K is is a polyhedral self-intersecting surface (Fig. 4). By construction, the graph G plays the role of the normal fan of K.

Model produced with: Maple 7 + JavaViewLib 3.95.000

Keywordsvirtual polytope; saddle surface; pointed embedding
MSC-2000 Classification52B70 (52B10, 52C20)


  1. A.D. Alexandrov: On uniqueness theorems for closed surfaces (Russian), Doklady Akad. Nauk SSSR 22 (1939), 99--102, .
  2. M. Knyazeva, G. Panina: An illustrated theory of hyperbolic virtual polytopes, CESJM, to appear (2008), .
  3. Y. Martinez-Maure: Contre-exemple \`a une caract\'erisation conjectur\'ee de la sph\`ere, C.R. Acad. Sci., Paris 332 (2001), 41--44, .
  4. G. Panina: New counterexamples to A.D. Alexandrov's uniqueness hypothesis, Advances in Geometry 5 (2005), 301--317.


240x240 preview

Submission information

Submitted: Thu Apr 17 10:06:36 CEST 2008.
Revised: Fri Mar 20 14:05:24 CET 2009, Tue Apr 21 10:05:42 CEST 2009, Thu May 7 11:21:58 CEST 2009.
Accepted: Thu Feb 18 12:13:46 CET 2010.

Authors' Addresses

Marina Knyazeva
Saint-Petersburg Institute for Informatics and Automation RAS
14 line V.O., 39

Gaiane Panina
Saint-Petersburg Institute for Informatics and Automation RAS
14 line V.O., 39