On H. Weyl and H. Minkowski polynomials.
On Hadwiger's problem on inner parallel bodies
We consider the problem of classifying the convex bodies in the 3-dimensional space depending on the differentiability of their associated quermassintegrals with respect to the one-parameter-depending family given by the inner/outer parallel bodies. It turns out that this problem is closely related to some behavior of the roots of the 3-dimensional Steiner polynomial.
On homogeneous coverings of Euclidean spaces.
On homotopy types of 2-dimensional polyhedra
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 Hyperplanes and Polytopes.
On hyperplanes and semispaces in max–min convex geometry
The concept of separation by hyperplanes and halfspaces is fundamental for convex geometry and its tropical (max-plus) analogue. However, analogous separation results in max-min convex geometry are based on semispaces. This paper answers the question which semispaces are hyperplanes and when it is possible to “classically” separate by hyperplanes in max-min convex geometry.
On Intersystolic Inequalities in Dimension 3.
On isoperimetric inequalities in Minkowski spaces.
On -neighbour packings and one-sided Hadwiger configurations.
On kissing numbers in dimensions 32 to 128.
On k-Sets in Arrangements of Curves and Surfaces.
On Lattice Points in Polyhedral Cross-Sections.
On lattice polytopes having interior lattice points.
On Lattices with Möbius Function ±1, 0.
On light subgraphs in plane graphs of minimum degree five
A subgraph of a plane graph is light if the sum of the degrees of the vertices of the subgraph in the graph is small. It is well known that a plane graph of minimum degree five contains light edges and light triangles. In this paper we show that every plane graph of minimum degree five contains also light stars and and a light 4-path P₄. The results obtained for and P₄ are best possible.
On Linear Sections of Lattice Packings.
On linked Jordan curves in
On local and global moduli of convexity
On local convexity of nonlinear mappings between Banach spaces
We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ɛ < c the image f(B ɛ(x)) of each ɛ-ball B ɛ(x) ⊂ U is convex. We give a lower bound on c via the second order Lipschitz constant Lip2(f), the Lipschitz-open constant Lipo(f) of f, and the 2-convexity number conv2(X) of the Banach space X.