Related papers: Approximate isoperimetry for convex polytopes
We study the problem of approximating the corner polyhedron using intersection cuts derived from families of lattice-free sets in $\mathbb{R}^n$. In particular, we look at the problem of characterizing families that approximate the corner…
We show that in dimensions $n \geq 12$, a non-flat complete gradient shrinking solitons with uniformly positive isotropic curvature (PIC) must be a quotient of either the round sphere $S^n$ or the cylinder $S^{n-1} \times \mathbb{R}$. We…
It is shown that if $C$ is an $n$-dimensional convex body then there is an affine image $\widetilde C$ of $C$ for which $${|\partial \widetilde C|\over |\widetilde C|^{n-1\over n}}$$ is no larger than the corresponding expression for a…
Every convex polygon with $n$ vertices is a linear projection of a higher-dimensional polytope with at most $147\,n^{2/3}$ facets.
Let $P\subset\mathbb{R}^n$ $(n\geq 3)$ be a convex polytope containing the origin in its interior. Let $\mbox{vol}_{n-2} \big( \mbox{relbd} ( P\cap\lbrace t\xi + \xi^\perp \rbrace ) \big)$ denote the $(n-2)$-dimensional volume of the…
A complete set of N+1 mutually unbiased bases (MUBs) forms a convex polytope in the N^2-1 dimensional space of NxN Hermitian matrices of unit trace. As a geometrical object such a polytope exists for all values of N, while it is unknown…
The isoperimetric problem with a density or weighting seeks to enclose prescribed weighted volume with minimum weighted perimeter. According to Chambers' recent proof of the log-convex density conjecture, for many densities on…
Facets of the convex hull of $n$ independent random vectors chosen uniformly at random from the unit sphere in $\mathbb{R}^d$ are studied. A particular focus is given on the height of the facets as well as the expected number of facets as…
We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also…
We show that there exist 0/1 polytopes in R^n with as many as (cn / (log n)^2)^(n/2) facets (or more), where c>0 is an absolute constant.
The main contribution of this paper is a new column-by-column method for the decomposition of generating functions of convex polyominoes suitable for enumeration with respect to various statistics including but not limited to interior…
It is a well known fact that in $\mathbb{R}^n$ a subset of minimal perimeter $L$ among all sets of a given volume is also a set of maximal volume among all sets of the same perimeter $L$. This is called the reciprocity principle for…
We consider the width $X_T(\omega)$ of a convex $n$-gon $T$ in the plane along the random direction $\omega\in\mathbb{R}/2\pi \mathbb{Z}$ and study its deviation rate: $$…
We prove the following comparison theorem for metrics with nonnegative scalar curvature, also known as the dihedral rigidity conjecture by Gromov: for $n\le 7$, if an $n$-dimensional prism has nonnegative scalar curvature and weakly mean…
Given an isotropic random vector $X$ with log-concave density in Euclidean space $\Real^n$, we study the concentration properties of $|X|$ on all scales, both above and below its expectation. We show in particular that: \[ \P(\abs{|X|…
The aim of this paper is to prove isoperimetric inequalities for simplices and polytopes with $d+2$ vertices in Euclidean, spherical and hyperbolic $d$-space. In particular, we find the minimal volume $d$-dimensional hyperbolic simplices…
Let $K$ be an isotropic symmetric convex body in ${\mathbb R}^n$. We show that a subspace $F\in G_{n,n-k}$ of codimension $k=\gamma n$, where $\gamma\in (1/\sqrt{n},1)$, satisfies $$K\cap F\subseteq \frac{c}{\gamma }\sqrt{n}L_K (B_2^n\cap…
According to Euler's relation any polytope P has as many faces of even dimension as it has faces of odd dimension. As a generalization of this fact one can compare the number of faces whose dimension is congruent to i modulo m with the…
The Upper Bound Theorem for convex polytopes implies that the $p$-th Betti number of the \v{C}ech complex of any set of $N$ points in $\mathbb R^d$ and any radius satisfies $\beta_{p} = O(N^{m})$, with $m = \min \{ p+1, \lceil d/2 \rceil…
In this paper we study convex subcomplexes of spherical buildings. We pay special attention to fixed point sets of type-preserving isometries of spherical buildings. This sets are also convex subcomplexes of the natural polyhedral structure…