Related papers: On a degeneracy ratio for bounded convex polytopes

We study geometrical aspects of entanglement, with the Hilbert--Schmidt norm defining the metric on the set of density matrices. We focus first on the simplest case of two two-level systems and show that a ``relativistic'' formulation leads…

Given a (finite) simplicial complex, we define its $i$-th Laplacian polytope as the convex hull of the columns of its $i$-th Laplacian matrix. This extends Laplacian simplices of finite simple graphs, as introduced by Braun and Meyer. After…

A group is said to be bounded if it has a finite diameter with respect to any bi-invariant metric. In the present paper we discuss boundedness of various groups of diffeomorphisms.

In this note we investigate the convex hull of those $n \times n$-permutation matrices that correspond to symmetries of a regular $n$-gon. We give the complete facet description. As an application, we show that this yields a Gorenstein…

We estimate the density of tubes around the algebraic variety of decomposable univariate polynomials over the real and the complex numbers.

The random beta polytope is defined as the convex hull of $n$ independent random points with the density proportional to $(1-\|x\|^2)^\beta$ on the $d$-dimensional unit ball, where $\beta>-1$ is a parameter. Similarly, the random beta'…

We study the regularity of the distance function to the boundary of a domain in $\mathbb{R}^n$, with respect to the Minkowski functional of a convex polytope. We obtain the regularity of the distance function in certain cases. We also…

We determine the Hausdorff limit-set of the Euclidean hypersurfaces with large $\lambda_1$ or small extrinsic radius. The result depends on the $L^p$ norm of the curvature that is assumed to be bounded a priori, with a critical behaviour…

We describe the deformation space of a solid torus with boundary modelled on convex ideal hyperbolic polyhedra. This deformation space is given by natural Gauss--Bonnet type inequalities on the dihedral angles. The result extends to solid…

We generalize the notion of expanded degenerations and pairs for a simple degeneration or smooth pair to the case of smooth Deligne-Mumford stacks. We then define stable quotients on the classifying stacks of expanded degenerations and…

We extend finding geometrically-significant preserved quantities by solving specific PDEs to the affine transformations and subgroups. This can be viewed not only as a purely geometrical problem but also as a subcase of finding physical…

A parametric manifold can be viewed as the manifold of orbits of a (regular) foliation of a manifold by means of a family of curves. If the foliation is hypersurface orthogonal, the parametric manifold is equivalent to the 1-parameter…

We advance a novel perspective of the entanglement issue that appeals to the Schlienz-Mahler measure [Phys. Rev. A 52, 4396 (1995)]. Related to it, we propose an criterium based on the consideration of convex subsets of quantum states. This…

We describe the first-order variations of the angles of Euclidean, spherical or hyperbolic polygons under infinitesimal deformations such that the lengths of the edges do not change. Using this description, we introduce a vector-valued…

The condition number of a smooth convex function, namely the ratio of its smoothness to strong convexity constants, is closely tied to fundamental properties of the function. In particular, the condition number of a quadratic convex…

We prove a lower bound theorem for the number of $k$-faces ($1\le k\le d-2$) in a $d$-dimensional polytope $P$ (or $d$-polytope) with up to $3d-1$ vertices. Previous lower bound theorems for $d$-polytopes with few vertices concern those…

In this paper, we consider the normality or the integer decomposition property (IDP, for short) for Minkowski sums of integral convex polytopes. We discuss some properties on the toric rings associated with Minkowski sums of integral convex…

We classify here combinatorially rigid simple polytopes with three facets more than their dimension.

We prove that for every convex body $K$ with the center of mass at the origin and every $\varepsilon\in \left(0,\frac{1}{2}\right)$, there exists a convex polytope $P$ with at most $e^{O(d)}\varepsilon^{-\frac{d-1}{2}}$ vertices such that…

We consider the set of points chosen randomly, independently and uniformly in the $d$-dimensional spherical layer. A set of points is called $1$-convex if all its points are vertices of the convex hull of this set. In \cite{3} an estimate…