Related papers: Quantitative non-embeddability theorems and metric…

An embeddability criterion for zero-dimensional metrizable topological spaces in zero-dimensional metrizable topological groups is given. A space which can be embedded as a closed subspace in a zero-dimensional metrizable group but is not…

We give an elementary proof of a compact embedding theorem in abstract Sobolev spaces. The result is first presented in a general context and later specialized to the case of degenerate Sobolev spaces defined with respect to nonnegative…

There are several Teichm\"uller spaces associated to a surface of infinite topological type, after the choice of a particular basepoint (a complex or a hyperbolic structure on the surface). These spaces include the quasiconformal…

Let $\mathsf{S}_1$ (the Schatten--von Neumann trace class) denote the Banach space of all compact linear operators $T:\ell_2\to \ell_2$ whose nuclear norm $\|T\|_{\mathsf{S}_1}=\sum_{j=1}^\infty\sigma_j(T)$ is finite, where…

We construct a doubling subset of $l_2$ which cannot be biLipschitz embedded in any finite dimensional Euclidean space. This answers a question of Lang and Plaut.

Although the Nash theorem solves the isometric embedding problem, matters are inherently more involved if one is further seeking an embedding that is well-behaved from the standpoint of submanifold geometry. More generally, consider a…

For $n\in \mathbb{N}$ consider the $n$-dimensional hypercube as equal to the vector space $\mathbb{F}_2^n$, where $\mathbb{F}_2$ is the field of size two. Endow $\mathbb{F}_2^n$ with the Hamming metric, i.e., with the metric induced by the…

Nontrivial isometric embeddings for flat metrics (i.e., those which are not just planes in the ambient space) can serve as useful tools in the description of gravity in the embedding gravity approach. Such embeddings can additionally be…

Non-Euclidean, or incompatible elasticity is an elastic theory for pre-stressed materials, which is based on a modeling of the elastic body as a Riemannian manifold. In this paper we derive a dimensionally-reduced model of the so-called…

We consider a large class of geodesic metric spaces, including Banach spaces, hyperbolic spaces and geodesic $\mathrm{CAT}(\kappa)$-spaces, and investigate the space of nonexpansive mappings on either a convex or a star-shaped subset in…

We investigate isoperimetric inequalities for Lipschitz 2-spheres in CAT(0) spaces, proving bounds on the volume of efficient null-homotopies. In one dimension lower, it is known that a quadratic inequality with a constant smaller than…

We study quasiisometric embeddings between finite-dimensional CAT(0) cube complexes. More specifically, we introduce geometric branching conditions under which flats in the domain, not necessarily of top rank, are mapped within finite…

Let $\H$ denote the discrete Heisenberg group, equipped with a word metric $d_W$ associated to some finite symmetric generating set. We show that if $(X,\|\cdot\|)$ is a $p$-convex Banach space then for any Lipschitz function $f:\H\to X$…

We prove that if an n-dimensional geodesically complete CAT(0) space has Tits boundary sufficiently close to the (n-1)-dimensional standard unit sphere, then it is bi-Lipschiz homeomorphic to the n-dimensional Euclidean space. As an…

In this paper, we introduce the notion of generalized $\epsilon$-stationarity for a class of nonconvex and nonsmooth composite minimization problems on compact Riemannian submanifold embedded in Euclidean space. To find a generalized…

The main result of the paper: Given any $\varepsilon>0$, every locally finite subset of $\ell_2$ admits a $(1+\varepsilon)$-bilipschitz embedding into an arbitrary infinite-dimensional Banach space. The result is based on two results which…

This article contains two rigidity type results for $\mathrm{SL}(n,\mathbb{Z})$ for large $n$ that share the same proof. Firstly, we prove that for every $p \in [1,\infty]$ different from $2$, the noncommutative $L^p$-space associated with…

Euclidean embeddings of data are fundamentally limited in their ability to capture latent semantic structures, which need not conform to Euclidean spatial assumptions. Here we consider an alternative, which embeds data as discrete…

We introduce a concept of an embedding of a quadratic space in an associative algebra. The general properties of such embeddings are analyzed by linking it to the Clifford algebra. Conversely, there isa simple description of the standard…

An inverse problem of elasticity of $n$ elastic inclusions embedded into an elastic half-plane is analyzed. The boundary of the half-plane is free of traction. The half-plane and the inclusions are subjected to antiplane shear, and the…