Related papers: Bi-Lipschitz homeomorphic subanalytic sets have bi…

Sampaio recently showed that bi-Lipschitz homeomorphic subanalytic sets have bi-Lipschitz homeomorphic tangent cones. The purpose of this note is to show that Sampaio's method works as well for (SSP) sets, that is, the above result is…

We prove that tangent cones of Lipschitz normally embedded sets are Lipschitz normally embedded. We also extend to real subanalytic sets the notion of reduced tangent cone and we show that subanalytic Lipschitz normally embedded sets have…

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 show that for every $k\ge 3$ there exist complex algebraic cones of dimension $k$ with isolated singularities, which are bi-Lipschitz and semi-algebraically equivalent but they have different degrees. We also prove that homeomorphic…

This paper deals with the problem of finding bi-Lipschitz behavior in non-degenerate Lipschitz maps between metric measure spaces. Specifically, we study maps from (subsets of) Ahlfors regular PI spaces into sub-Riemannian Carnot groups. We…

We show that a weighted homogeneous complex surface singularity is metrically conical (i.e., bi-Lipschitz equivalent to a metric cone) only if its two lowest weights are equal. We also give an example of a pair of weighted homogeneous…

Let $\sigma_q : \mathbb{R}^q \to {\bf S}^q \setminus N_q$ be the inverse of the stereographic projection with centre the north pole $N_q$. Let $W_i$ be a closed subset of $\mathbb{R}^{q_i}$, for $i=1,2$. Let $\Phi:W_1 \to W_2$ be a…

We prove that any analytic set in $\C^n$ with a unique tangent cone at infinity is an algebraic set. We prove that the degree of a complex algebraic set in $\C^n$, which is Lipschitz normally embedded at infinity, is equal to the degree of…

We prove that any complex analytic set in $\mathbb{C}^n$ which is Lipschitz normally embedded at infinity and has tangent cone at infinity that is a linear subspace of $\mathbb{C}^n$ must be an affine linear subspace of $\mathbb{C}^n$…

We show that a topological symplectic manifold has a canonically associated bi-Lipschitz structure. As a corollary, we obtain the first examples of non-existence and non-uniqueness for topological symplectic structures. Our arguments hold…

We introduce cone bilipschitz equivalences between metric spaces. These are maps, more general than quasi-isometries, that induce a bilipschitz homeomorphism between asymptotic cones. Non-trivial examples appear in the context of Lie…

In this article, we study the Lipschitz Geometry at infinity of complex analytic sets and we obtain results on algebraicity of analytic sets and on Bernstein's problem. Moser's Bernstein Theorem says that a minimal hypersurface which is a…

In this paper, we extend the result of arXiv:2409.13662 by showing that the set on which every pseudotangent is obtained on a Lipschitz curve can be any compact, uniformly disconnected set in Euclidean space which admits any Lipschitz…

In this paper we investigate the behaviour of the geometric directional bundles, associated to arbitrary subsets in R^n, under bi-Lipschitz homeomorphisms, and give conditions under which their bi-Lipschitz type is preserved. The most…

We study invariance of multiplicity of complex analytic germs and degree of complex affine sets under outer bi-Lipschitz transformations (outer bi-Lipschitz homeomorphims of germs in the first case and outer bi-Lipschitz homeomorphims at…

We prove some epsilon regularity results for n-dimensional minimal two-valued Lipschitz graphs. The main theorems imply uniqueness of tangent cones and regularity of the singular set in a neighbourhood of any point at which at least one…

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…

We show that bi-Lipschitz conjugacies between non singular one dimensional systems are forced to be smooth, at least in the minimal (and ergodic) case. This is however far from being true in the non minimal case. These results clarify a…

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 introduce the notion of local quasi-isometry for metric germs and prove that two definable germs are quasi-isometric if and only if their tangent cones are bi-Lipschitz homeomorphic. Since bi-Lipschitz equivalence is a…