Related papers: A non-injective Assouad-type theorem with sharp di…

We give a new approach to the infinitesimal structure of Lipschitz maps into L^1. As a first application, we give an alternative proof of the main theorem from an earlier paper, that the Heisenberg group does not admit a bi-Lipschitz…

We introduce and study bi-Lipschitz-invariant dimensions that range between the box and Assouad dimensions. The quasi-Assouad dimensions and $\theta$-spectrum are other special examples of these intermediate dimensions. These dimensions are…

Let $M$ be a closed smooth connected spin manifold of even dimension $n$, let $g$ be a Riemannian metric of regularity $W^{1,p}$, $p > n$, on $M$ whose distributional scalar curvature in the sense of Lee-LeFloch is bounded below by…

Hurewicz's dimension-raising theorem states that for every n-to-1 map f : X \to Y, dim Y =< dim X + n holds. In this paper we introduce a new notion of finite-to-one like map in a large scale setting. Using this notion we formulate a…

We solve the following three questions concerning surjective linear isometries between spaces of Lipschitz functions $\mathrm{Lip}(X,E)$ and $\mathrm{Lip}(Y,F)$, for strictly convex normed spaces $E$ and $F$ and metric spaces $X$ and $Y$:…

We study Lipschitz differentiability spaces, a class of metric measure spaces introduced by Cheeger. We show that if an Ahlfors regular Lipschitz differentiability space has charts of maximal dimension, then, at almost every point, all its…

Given a finite collection $\{X_i\}_{i\in I}$ of metric spaces, each of which has finite Nagata dimension and Lipschitz free space isomorphic to $L^1$, we prove that their union has Lipschitz free space isomorphic to $L^1$. The short proof…

A well-known result by Lindenstrauss is that any two-dimensional normed space can be isometrically imbedded into $L_1(0,1)$. We provide an explicit form of a such an imbedding. The proof is elementary and self-contained. Applications are…

We introduce the notion of a two-scale branching function associated with an arbitrary metric space, which encodes the lower and upper box dimensions as well as the Assouad spectrum. If the metric space is quasi-doubling, this function is…

A metric measure space is said to be Carnot-rectifiable if it can be covered up to a null set by countably many biLipschitz images of compact sets of a fixed Carnot group. In this paper, we give several characterisations of such notion of…

Let $X$ be an $n$-point subset of a Euclidean space and $0 < a < 1$. The classical theorem of Schoenberg implies that the snowflake space $X^a$ can be isometrically embedded into Euclidean space. In the paper we show that points in the…

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…

We give a short proof that any non-zero Euclidean space has a compact subset of Hausdorff dimension one that contains a differentiability point of every real-valued Lipschitz function defined on the space.

As our main theorem, we prove that a Lipschitz map from a compact Riemannian manifold $M$ into a Riemannian manifold $N$ admits a smooth approximation via immersions if the map has no singular points on $M$ in the sense of F.H. Clarke,…

We prove a Lipschitz-Volume rigidity theorem in Alexandrov geometry, that is, if a 1-Lipschitz map $f\colon X=\amalg X_\ell\to Y$ between Alexandrov spaces preserves volume, then it is a path isometry and an isometry when restricted to the…

We prove that if a metric space $X$ has Nagata dimension zero with constant $c$, then there exists a dense subset of $X$ that is $8c$-bilipschitz equivalent to a weighted tree. The factor $8$ is the best possible if $c=1$, that is, if $X$…

Motivated by the Lipschitz rigidity problem in scalar curvature geometry, we prove that if a closed smooth spin manifold admits a distance decreasing continuous map of non-zero degree to a sphere, then either the scalar curvature is…

We study the fine scaling properties of sets satisfying various weak forms of invariance. For general attractors of possibly overlapping bi-Lipschitz iterated function systems, we establish that the Assouad dimension is given by the…

For Lipschitz maps between a metric measure space and a metric space, combining the ideas of Kirchheim's metric differentiability and Cheeger's differentiable structures leads to a Rademacher-type theorem for a notion of metric…

Spacetimes obtained by dimensional reduction along lattices containing a lightlike direction can admit semigroup extensions of their isometry groups. We show by concrete examples that such a semigroup can exhibit a natural order, which in…