Related papers: The Geometric Structure of Max-Plus Hemispaces

A complete set of N+1 mutually unbiased bases (MUBs) exists in Hilbert spaces of dimension N = p^k, where p is a prime number. They mesh naturally with finite affine planes of order N, that exist when N = p^k. The existence of MUBs for…

It is well-known that the sequence of iterations of the composition of projections onto closed affine subspaces converges linearly to the projection onto the intersection of the affine subspaces when the sum of the corresponding linear…

We determine the maximal hyperplane sections of the regular $n$-simplex, if the distance of the hyperplane to the centroid is fairly large, i.e. larger than the distance of the centroid to the midpoint of edges. Similar results for the…

We are generalizing to higher dimensions the Bavard-Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex d-dimensional polyhedra with fixed directions of facet normals has a…

We present a geometrical analysis on the completely positive programming reformulation of quadratic optimization problems and its extension to polynomial optimization problems with a class of geometrically defined nonconvex conic programs…

The notion of max-plus convex subset of Euclidean space can be naturally extended to other linear spaces. The aim of this paper is to describe the topology of hyperspaces of max-plus convex subsets of Tychonov powers $\mathbb R^\tau$ of the…

This paper is concerned with the completeness (with respect to the centroaffine metric) of hyperbolic centroaffine hypersurfaces which are closed in the ambient vector space. We show that completeness holds under generic regularity…

We characterize embedded $\C^1$ hypersurfaces of $\R^n$ as the only locally closed sets with continuously varying flat tangent cones whose measure-theoretic-multiplicity is at most $m<3/2$. It follows then that any (topological)…

Let $P \subset \mathbb{R}^{d}$ be a closed convex cone. Assume that $P$ is pointed, i.e. the intersection $P \cap -P=\{0\}$ and $P$ is spanning, i.e. $P-P=\mathbb{R}^{d}$. Denote the interior of $P$ by $\Omega$. Let $E$ be a product system…

We study the structure of the set of all possible affine hyperplane sections of a convex polytope. We present two different cell decompositions of this set, induced by hyperplane arrangements. Using our decomposition, we bound the number of…

In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $\Sigma$ results in a properly convex projective structure on $M$ with finite volume. We also discuss various geometric…

The convexity number of a set $X \subset \mathbb{R}^2$ is the minimum number of convex subsets required to cover it. We study the following question: what is the largest possible convexity number $f(n)$ of $\mathbb{R}^2 \setminus S$, where…

This paper is devoted to the classification problems concerning extended deformations of convex polyhedra and real hyperplane arrangements in the following senses: combinatorial equivalence of face posets, normal equivalence on normal fans…

Y. Benoist proved that if a closed three-manifold M admits an indecomposable convex real projective structure, then M is topologically the union along tori and Klein bottles of finitely many sub-manifolds each of which admits a complete…

The theory of geometric structures on a surface with nonempty boundary can be developed by using a decomposition of such a surface into hexagons, in the same way as the theory of geometric structures on a surface without boundary is…

Using combinatorial methods, we derive several formulas for the volume of convex bodies obtained by intersecting a unit hypercube with a halfspace, or with a hyperplane of codimension 1, or with a flat defined by two parallel hyperplanes.…

Optimizing an implicational base of a closure system consists in turning this implicational base into an equivalent one with premises and conclusions as small as possible. This task is known to be hard in general but tractable for a number…

We study the slices or sections of a convex polytope by affine hyperplanes. We present results on two key problems: First, we provide tight bounds on the maximum number of vertices attainable by a hyperplane slice of $d$-polytope (a sort of…

Let $M$ be a geometrically finite acylindrical hyperbolic 3-manifold and let $M^*$ denote the interior of the convex core of M. We show that any geodesic plane in $M^*$ is either closed or dense, and that there are only countably many…

In this article we construct many examples of properly convex irreducible domains divided by Zariski dense relatively hyperbolic groups in every dimension at least 3. This answers a question of Benoist. Relative hyperbolicity and non-strict…