Related papers: Duality of non-exposed faces
We study hyperbolic polyhedral surfaces with faces isometric to regular hyperbolic polygons satisfying that the total angles at vertices are at least $2\pi.$ The combinatorial information of these surfaces is shown to be identified with…
Iterated planar contact manifolds are a generalization of three dimensional planar contact manifolds to higher dimensions. We study some basic topological properties of iterated planar contact manifolds and discuss several examples and…
We give new bounds on the Erdos-Szekeres theorems for convex bodies of Bisztriczky and Fejes Toth and of Pach and Toth. We derive them from a combinatorial characterization of convex position of a family of planar convex bodies. This…
A convex body is unconditional if it is symmetric with respect to reflections in all coordinate hyperplanes. In this paper, we investigate unconditional lattice polytopes with respect to geometric, combinatorial, and algebraic properties.…
Convex co-compact 3-dimensional hyperbolic manifolds are uniquely determined by the pleating measured lamination on the boundary of their convex core.
We study focal surfaces of (wave) fronts associated to unbounded principal curvatures near non-degenerate singular points of initial fronts. We give characterizations of singularities of those focal surfaces in terms of types of…
We study the number of facets of the convex hull of n independent standard Gaussian points in d-dimensional Euclidean space. In particular, we are interested in the expected number of facets when the dimension is allowed to grow with the…
The structure of cones of positive and k-positive maps acting on a finite-dimensional Hilbert space is investigated. Special emphasis is given to their duality relations to the sets of superpositive and k-superpositive maps. We characterize…
We introduce complex intersection bodies and show that their properties and applications are similar to those of their real counterparts. In particular, we generalize Busemann's theorem to the complex case by proving that complex…
An antinorm is a concave analogue of a norm. In contrast to norms, antinorms are not defined on the entire space $R^d$ but on a cone $K\subset R^d$. They are applied in the matrix analysis, optimal control, and dynamical systems. Their…
{We show in this paper that two normal elliptic sections through every point of the boundary of a smooth convex body essentially characterize an ellipsoid and furthermore, that four different pairwise non-tangent elliptic sections through…
In the present paper, we will show that three apparently disjoint objects: Galois representations arising from twenty-seven lines on a cubic surface (number theory and arithmetic algebraic geometry), Picard modular forms (automorphic…
A permutation polytope is the convex hull of a group of permutation matrices. In this paper we investigate the combinatorics of permutation polytopes and their faces. As applications we completely classify permutation polytopes in…
Building sets were introduced in the study of wonderful compactifications of hyperplane arrangement complements and were later generalized to finite meet-semilattices. Convex geometries, the duals of antimatroids, offer a robust…
We introduce a relation of block conjugacy for irreducible toral automorphism, and prove that block conjugacy is equivalent to weak equivalence of the ideals associated to the automorphisms. We characterize when block conjugate…
We investigate the topological structures of Galois covers of a union of two Zappatic surfaces of type $R_k$. We prove that the Galois covers of such surfaces are simply-connected surfaces of general type. We also compute their Chern…
We define and study a new family of polytopes which are formed as convex hulls of partial alternating sign matrices. We determine the inequality descriptions, number of facets, and face lattices of these polytopes. We also study partial…
Using the orthogonal connectedness, we introduce the notion of orthogonal decomposability of convex polytopes and study it in the case of Platonic and Archimedean solids. While doing so, we also encounter polytopes which are not…
We study linearizability of actions of finite groups on cubic threefolds with non-isolated singularities.
Suppose S is a planar set. Two points a,b in S 'see each other' via S if [a,b] is included in S . F. Valentine proved in 1957 that if S is closed, and if for every three points of S, at least two see each other via S, then S is a union of…