Related papers: Normal polytopes: between discrete, continuous, an…
For a given convex body K in $R^d$, a random polytope $K^{(n)}$ is defined (essentially) as the intersection of $n$ independent closed halfspaces containing $K$ and having an isotropic and (in a specified sense) uniform distribution. We…
This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part…
This paper introduces the notion of an unravelled abstract regular polytope, and proves that $\SL_3(q) \rtimes <t>$, where $t$ is the transpose inverse automorphism of $\SL_3(q)$, possesses such polytopes for various congruences of $q$. A…
A mostly expository account of old questions about the relationship between polyhedra and topological manifolds. Topics are old topological results, new gauge theory results (with speculations about next directions), and history of the…
We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for polytopes defined by integer constraints in terms of the height of the integers and certain subdeterminants of the constraint…
These are lecture notes supporting a minicourse taught at the Summer School in Total Positivity and Quantum Field Theory at CMSA Harvard in June 2025. We give an introduction to positive geometries and their canonical forms. We present the…
We study the vertices of the polytopes of all affine maps (a.k.a. hom-polytopes) between higher dimensional simplices, cubes, and crosspolytopes. Systematic study of general hom-polytopes was initiated in [3]. The study of such vertices is…
With respect to earlier investigations, the theory of multi-component, concentric, copolar, axisymmetric, rigidly rotating polytropes is improved and extended, including subsystems with nonzero density on the boundary and subsystems with…
Abstract polytopes are combinatorial structures with distinctive geometric, algebraic, or topological characteristics, that generalize (the face lattice of) traditional polyhedra, polytopes or tessellations. Most research has focused on…
This paper develops asymptotic methods to count faces of random high-dimensional polytopes. Beyond its intrinsic interest, our conclusions have surprising implications - in statistics, probability, information theory, and signal processing…
We study the Hausdorff distance between a random polytope, defined as the convex hull of i.i.d. random points, and the convex hull of the support of their distribution. As particular examples, we consider uniform distributions on convex…
Motivated by the analysis of the performance of the simplex method we study the behavior of families of pivot rules of linear programs. We introduce normalized-weight pivot rules which are fundamental for the following reasons: First, they…
The random polytope $K_n$, defined as the convex hull of $n$ points chosen uniformly at random on the boundary of a smooth convex body, is considered. Proofs for lower and upper variance bounds, strong laws of large numbers and central…
Let $X_1,\ldots,X_N$, $N>n$, be independent random points in $\mathbb{R}^n$, distributed according to the so-called beta or beta-prime distribution, respectively. We establish threshold phenomena for the volume, intrinsic volumes, or more…
This survey contains a selection of topics unified by the concept of positive semi-definiteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or…
A classification of discrete polymatroids whose independence polytopes are reflexive will be presented.
We construct, for each convex polytope, possibly nonrational and nonsimple, a family of compact spaces that are stratified by quasifolds, i.e. each of these spaces is a collection of quasifolds glued together in an suitable way. A quasifold…
Abstract polytopes generalize the classical notion of convex polytopes to more general combinatorial structures. The most studied ones are regular and chiral polytopes, as it is well-known, they can be constructed as coset geometries from…
We review a collection of models of random simplicial complexes together with some of the most exciting phenomena related to them. We do not attempt to cover all existing models, but try to focus on those for which many important results…
The present paper aims to survey known results and to point out the wealth of rather important open problems that are out there.