Related papers: Width of l^p balls

We discuss various questions of the following kind: for a continuous map $X \to Y$ from a compact metric space to a simplicial complex, can one guarantee the existence of a fiber large in the sense of Urysohn width? The $d$-width measures…

Graphs on integer points of polytopes whose edges come from a set of allowed differences are studied. It is shown that any simple graph can be embedded in that way. The minimal dimension of such a representation is the fiber dimension of…

We study symplectic embeddings of ellipsoids into balls. In the main construction, we show that a given embedding of 2m-dimensional ellipsoids can be suspended to embeddings of ellipsoids in any higher dimension. In dimension 6,s if the…

Alexandrov's Theorem states that every metric with the global topology and local geometry required of a convex polyhedron is in fact the intrinsic metric of a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a…

The dimension of a graph $G$ is the smallest $d$ for which its vertices can be embedded in $d$-dimensional Euclidean space in the sense that the distances between endpoints of edges equal $1$ (but there may be other unit distances).…

We study the embedding $\text{id}: \ell_p^b(\ell_q^d) \to \ell_r^b(\ell_u^d)$ and prove matching bounds for the entropy numbers $e_k(\text{id})$ provided that $0<p<r\leq \infty$ and $0<q\leq u\leq \infty$. Based on this finding, we…

Let $\mathcal{M}$ be a smooth submanifold of $\mathbb{R}^n$ equipped with the Euclidean (chordal) metric. This note considers the smallest dimension $m$ for which there exists a bi-Lipschitz function $f: \mathcal{M} \mapsto \mathbb{R}^m$…

We study the size (or volume) of balls in the metric space of permutations, $S_n$, under the infinity metric. We focus on the regime of balls with radius $r = \rho \cdot (n\!-\!1)$, $\rho \in [0,1]$, i.e., a radius that is a constant…

In his work on singularities, expanders and topology of maps, Gromov showed, using isoperimetric inequalities in graded algebras, that every real valued map on the $n$-torus admits a fibre whose homological size is bounded below by some…

We consider the problem of estimating the distance between two bodies of volume $\varepsilon$ located inside a $n$-dimensional ball $U$ of unit volume for $n\to\infty$. Let $A$ be a closed set with a smooth boundary of the volume…

A spectrahedron is a set defined by a linear matrix inequality. Given a spectrahedron we are interested in the question of the smallest possible size $r$ of the matrices in the description by linear matrix inequalities. We show that for the…

Let $\mathcal{P}$ be an $n$-point subset of Euclidean space and $d\geq 3$ be an integer. In this paper we study the following question: What is the smallest (normalized) relative change of the volume of subsets of $\mathcal{P}$ when it is…

We give sufficient conditions for a metric space to bilipschitz embed in L_1. In particular, if X is a length space and there is a Lipschitz map u:X--->R such that for every interval I in R, the connected components of the inverse image…

Average distance between two points in a unit-volume body $K \subset \mathbb{R}^n$ tends to infinity as $n \to \infty$. However, for two small subsets of volume $\varepsilon > 0$ the situation is different. For unit-volume cubes and…

We determine upper asymptotic estimates of Kolmogorov and linear $n$-widths of unit balls in Sobolev and Besov norms in $L_{p}$-spaces on compact Riemannian manifolds. The proofs rely on estimates for the near-diagonal localization of the…

In this paper we consider piecewise linear (pl) isometric embeddings of Euclidean polyhedra into Euclidean space. A Euclidean polyhedron is just a metric space $\mathcal{P}$ which admits a triangulation $\mathcal{T}$ such that each…

We prove a generalized isoperimetric inequality for a domain diffeomorphic to a sphere that replaces filling volume with $k$-dilation. Suppose $U$ is an open set in $\mathbb{R}^n$ diffeomorphic to a Euclidean $n$-ball. We show that in…

We present a surprisingly short proof that for any continuous map $f : \mathbb{R}^n \rightarrow \mathbb{R}^m$, if $n>m$, then there exists no bound on the diameter of fibers of $f$. Moreover, we show that when $m=1$, the union of small…

Let $X$ be a set of $n$ points of norm at most $1$ in the Euclidean space $R^k$, and suppose $\varepsilon>0$. An $\varepsilon$-distance sketch for $X$ is a data structure that, given any two points of $X$ enables one to recover the square…

The notion of the Urysohn $d$-width measures to what extent a metric space can be approximated by a $d$-dimensional simplicial complex. We investigate how local Urysohn width bounds on a riemannian manifold affect its global width. We bound…