Objects that Cannot Be Taken Apart with Two Hands.
On a Carathéodory's conjecture on umbilics : representing ovaloids
On a Conjectured Characterization of the Sphere.
On affine subspaces that illuminate a convex set.
On faces of a convex polyhedron in with a small number of sides.
On hyperbolic virtual polytopes and hyperbolic fans
Hyperbolic virtual polytopes arose originally as polytopal versions of counterexamples to the following A.D.Alexandrov’s uniqueness conjecture: Let K ⊂ ℝ3 be a smooth convex body. If for a constant C, at every point of ∂K, we have R 1 ≤ C ≤ R 2 then K is a ball. (R 1 and R 2 stand for the principal curvature radii of ∂K.) This paper gives a new (in comparison with the previous construction by Y.Martinez-Maure and by G.Panina) series of counterexamples to the conjecture. In particular, a hyperbolic...
On optimal route planning evading cubes in the three space.
On the affine convexity of convex curves and hypersurfaces.
On the Connected Components of the Space of Line Transversals to a Family of Convex Sets.
On the Union of Jordan Regions and Collision-Free Translational Motion Amidst Polygonal Obstacles.