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.
Bounded linear regularity, strong CHIP, and CHIP are distinct properties.
Circles holding a regular triangular prism.
Circumscribing constant-width bodies with polytopes.
Convex Disks Can Cover Their Shadow.
Countably convex sets
We investigate countably convex subsets of Banach spaces. A subset of a linear space is countably convex if it can be represented as a countable union of convex sets. A known sufficient condition for countable convexity of an arbitrary subset of a separable normed space is that it does not contain a semi-clique [9]. A semi-clique in a set S is a subset P ⊆ S so that for every x ∈ P and open neighborhood u of x there exists a finite set X ⊆ P ∩ u such that conv(X) ⊈ S. For closed sets this condition...
Curvatures of conflict surfaces in Euclidean 3-space
This article extends to three dimensions results on the curvature of the conflict curve for pairs of convex sets of the plane, established by Siersma [3]. In the present case a conflict surface arises, equidistant from the given convex sets. The Gaussian, mean curvatures and the location of umbilic points on the conflict surface are determined here. Initial results on the Darbouxian type of umbilic points on conflict surfaces are also established. The results are expressed in terms of the principal...
Eine Charaketrisierung der Sphäre im E3.
Erdős-Mordell inequality for space n-gons.
How to cage an egg.
Inner parallel curves and domains in the equiaffine geometry.
Konvexe Körper approximierende Polytopklassen.