A new class of stuck unknots in .
A normed space with the Beckman-Quarles property.
A short proof of rigidity of convex polytopes.
Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes
We present a constructive proof of Alexandrov’s theorem on the existence of a convex polytope with a given metric on the boundary. The polytope is obtained by deforming certain generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a connection with weighted Delaunay triangulations of polyhedral surfaces. The existence of the deformation follows from the non-degeneracy of the Hessian of the total scalar curvature of generalized convex...
Around the proof of the Legendre-Cauchy lemma on convex polygons.
Configuration spaces of weighted graphs in high dimensional Euclidean spaces.
Flexible cross-polytopes in the Euclidean 4-space.
Higher-Order Rigidity ? What Is the Proper Definition?
Iterations of concave maps, the Perron-Frobenius theory, and applications to circle packings.
La conjecture des soufflets
On sait depuis les travaux de Bricard et de Connelly qu’il existe dans l’espace euclidien des polyèdres (non convexes) qui sont flexibles : on peut les déformer continûment sans changer la forme de leurs faces. La conjecture des soufflets affirme que le volume interieur de ces polyèdres est constant au cours de la déformation. Elle a été démontrée récemment par I. Sabitov, qui a pour cela utilisé des outils algébriques inattendus dans ce contexte.
On the notion of the combinatorial -parametric property for polyhedra.
On the rigidity of one-dimensional systems of contraction similitudes.
Rigidity and flexibility of virtual polytopes
All 3-dimensional convex polytopes are known to be rigid. Still their Minkowski differences (virtual polytopes) can be flexible with any finite freedom degree. We derive some sufficient rigidity conditions for virtual polytopes and present some examples of flexible ones. For example, Bricard's first and second flexible octahedra can be supplied by the structure of a virtual polytope.
Rigidity and the Alexandrov-Fenchel Inequality.
Rigidity of infinite disk patterns.
Spaces of stresses, projections and parallel drawings for spherical polyhedra.
Starre, kippende, wackelige und bewegliche Gelenkvierecke im Raum.
Symmetric Laman theorems for the groups and .
The Bellows conjecture.
Transformationen von Polyedern in Polyederketten.