A short proof of rigidity of convex polytopes.
A simpler construction of volume polynomials for a polyhedron.
A symbolic shortest path algorithm for computing subgame-perfect Nash equilibria
Consider games where players wish to minimize the cost to reach some state. A subgame-perfect Nash equilibrium can be regarded as a collection of optimal paths on such games. Similarly, the well-known state-labeling algorithm used in model checking can be viewed as computing optimal paths on a Kripke structure, where each path has a minimum number of transitions. We exploit these similarities in a common generalization of extensive games and Kripke structures that we name “graph games”. By extending...
A theorem on nonexistence of a certain type of nearly regular cell-decompositions of the sphere
Acknowledgment of Priority Concerning: Polytopes that Fill R" and Scissors Congruence.
Alexander duality in subdivisions of Lawrence 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...
Algebraic shifting and sequentially Cohen-Macaulay simplicial complexes.
Algorithms for investigating optimality of cone triangulation for a polyhedron
Algorithms for Triangulating Polyhedra Into a Small Number of Tethraedra
An Ehrhart series formula for reflexive polytopes.
An example of a flexible polyhedron with nonconstant volume in the spherical space.
An illustrated theory of hyperbolic virtual polytopes
The paper gives an illustrated introduction to the theory of hyperbolic virtual polytopes and related counterexamples to A.D. Alexandrov’s conjecture.
An inequality concerning edges of minor weight in convex 3-polytopes
Let be the number of edges in a convex 3-polytope joining the vertices of degree i with the vertices of degree j. We prove that for every convex 3-polytope there is ; moreover, each coefficient is the best possible. This result brings a final answer to the conjecture raised by B. Grünbaum in 1973.
An Infinite Family of Minor-Minimal Nonrealizable 3-Chirotopes.
An integral geometric theorem for simple valuations.
An Interactive Creation of Polyhedra Stellations with Various Symmetries
An intrinsic characterization of foldings of euclidean space
An isoperimetric problem for tetrahedra.
Andreev’s Theorem on hyperbolic polyhedra
In 1970, E.M.Andreev published a classification of all three-dimensional compact hyperbolic polyhedra (other than tetrahedra) having non-obtuse dihedral angles. Given a combinatorial description of a polyhedron, , Andreev’s Theorem provides five classes of linear inequalities, depending on , for the dihedral angles, which are necessary and sufficient conditions for the existence of a hyperbolic polyhedron realizing with the assigned dihedral angles. Andreev’s Theorem also shows that the resulting...