Related papers: The kissing dimension of subanalytic sets is prese…

We introduce a notion of graph homeomorphisms which uses the concept of dimension and homotopy for graphs. It preserves the dimension of a subbasis, cohomology and Euler characteristic. Connectivity and homotopy look as in classical…

For any metric space $X$, finite subset spaces of $X$ provide a sequence of isometric embeddings $X=X(1)\subset X(2)\subset\cdots$. The existence of Lipschitz retractions $r_n\colon X(n)\to X(n-1)$ depends on the geometry of $X$ in a subtle…

Suppose A is an open subset of a Carnot group G, where G has a discrete analogue, and H is another Carnot group. We show that a Lipschitz function from A to H whose image has positive Hausdorff measure in the appropriate dimension is…

Let n>2 and X be a Banach space of dimension strictly greater than n. We show there exists a directionally porous set P in X for which the set of C^1 surfaces of dimension n meeting P in positive measure is not meager. If X is separable…

Let $U$ be an open relatively compact subanalytic subset of a real analytic manifold. We show that there exists a finite linear covering (in the sense of Guillermou and Schapira) of $U$ by subanalytic open subsets of $U$ homeomorphic to a…

A question of B. Teissier, inspired by a previous problem of R. Thom, asks whether for any germ of complex analytic hypersurface there exists a germ of complex algebraic hypersurface with the same topological type. Up to now only the case…

Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. These two metrics…

In this short note, we show that, in any given metric space, every Lipschitz open-map image of every subset of a given metric space whose boundary is Hausdorff-null is Hausdorff-measurable with respect to the same dimension. The main…

We present here basic results in Lipschitz Geometry of semialgebraic surface germs. Although bi-Lipschitz classification problem of surface germs with respect to the inner metric was solved long ago, classification with respect to the outer…

We generalize a bi-Lipschitz extension result of David and Semmes from Euclidean spaces to complete metric measure spaces with controlled geometry (Ahlfors regularity and supporting a Poincar\'e inequality). In particular, we find sharp…

We show that for the base two expansion \[ x=\sum_{i=1}^{\infty}2^{-(d_{1}(x)+d_{2}(x)+\dots+d_{i}(x))}\] with $x\in(0,1]$ and $d_{i}(x)\in\mathbb{N}$ the set $A=\{x|\lim_{i\to\infty}d_{i}(x)=\infty\}$ has Hausdorff dimension zero, this is…

Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. A complex analytic…

In this article, we prove that for a definable set in an o-minimal structure with connected link (at 0 or infinity), the inner distance of the link is equivalent to the inner distance of the set restricted to the link. With this result, we…

Let $N$ be an $n$-dimensional compact riemannian manifold, with $n\geq 2$. In this paper, we prove that for any $\alpha\in [0,n]$, the set consisting of homeomorphisms on $N$ with lower and upper metric mean dimensions equal to $\alpha$ is…

We compare, for L a Lie ring over the integers, its lower central series (\gamma_n(L))_{n>0} and its dimension series defined by \delta_n(L):=L\cap \varpi^n(L) in the universal enveloping algebra of L. We show that \gamma_n(L)=\delta_n(L)…

We give conditions on a general family $P_{\lambda}:\R^n\to\R^m, \lambda \in \Lambda,$ of orthogonal projections which guarantee that the Hausdorff dimension formula $\dim A\cap P_{\lambda}^{-1}\{u\}=s-m$ holds generically for measurable…

These notes constitute a survey on the geometric properties of globally subanalytic sets. We start with their definition and some fundamental results such as Gabrielov's Complement Theorem or existence of cell decompositions. We then give…

In this paper, we prove metric analogues, in any dimension and in any co-dimension, of the famous Theorem of Mumford on smoothness of normal surfaces and the beautiful Theorem of Ramanujam that gives a topological characterization of…

We investigate the influence that $s$-dimensional lower and upper Hausdorff densities have on the geometry of a Radon measure in $\mathbb{R}^n$ when $s$ is a real number between $0$ and $n$. This topic in geometric measure theory has been…

We introduce a technique that uses projection properties of fractal percolation to establish dimension conservation results for sections of deterministic self-similar sets. For example, let $K$ be a self-similar subset of $\mathbb{R}^2$…