A characterization of spheres among compact 3-bodies satisfying Crofton's theorem
A characterization of spheres among convex 3-bodies
A Linear-Time Algorithm for Computing the Voronoi Diagram of a Convex Polygon.
A theorem on nonexistence of a certain type of nearly regular cell-decompositions of the sphere
A typical convex surface contains no closed geodesic.
An alternative proof of Petty's theorem on equilateral sets
The main goal of this paper is to provide an alternative proof of the following theorem of Petty: in a normed space of dimension at least three, every 3-element equilateral set can be extended to a 4-element equilateral set. Our approach is based on the result of Kramer and Németh about inscribing a simplex into a convex body. To prove the theorem of Petty, we shall also establish that for any three points in a normed plane, forming an equilateral triangle of side p, there exists a fourth point,...
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 isoperimetric problem for tetrahedra.
Asphericity of shadows of a convex body.