Related papers: On the bi-Lipschitz contact equivalence of plane c…
We discuss the problem of deciding when a metrisable topological group $G$ has a canonically defined local Lipschitz geometry. This naturally leads to the concept of minimal metrics on $G$, that we characterise intrinsically in terms of a…
A classification of upper semicontinuous, translation and dually epi-translation invariant valuations is established on the space of convex Lipschitz function on $\mathbb{R}$ with compact domain.
We introduce a natural definition of $L^p$-convergence of maps, $p \ge 1$, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a…
It is shown that, generically, a positive continuous functions cannot be written as the derivative of a bi-Lipschitz plane homeomorphism. The same is proved for generic bounded functions.
We prove that the Hilbert geometry of a product of convex sets is bi-lipschitz equivalent the direct product of their respective Hilbert geometries. We also prove that the volume entropy is additive with respect to product and that…
The aim of this paper to introduce the reader to a recent point of view on the Lipschitz classifications of complex singularities. It presents the complete classification of Lipschitz geometry of complex plane curves singularities and in…
A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. We study the geometry of these trees in two directions. First, we construct a…
We characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined on compact Riemannian manifolds. Through…
We interpret the construction of relative Cuntz-Pimsner algebras of correspondences in terms of the correspondence bicategory, as a reflector into a certain sub-bicategory. This generalises a previous characterisation of absolute…
We prove that any correspondence (multi-function) mapping a metric space into a Banach space that satisfies a certain pointwise Lipschitz condition, always has a continuous selection that is pointwise Lipschitz on a dense set of its domain.…
We give sufficient conditions to guarantee that if two self-conformal sets E and F have Lipschitz equivalent subsets of positive measure, then there is a bilipschitz map of E into, or onto, F.
We prove that Ahlfors 2-regular quasisymmetric images of the Euclidean plane are bi-Lipschitz images of the plane if and only if they are uniformly bi-Lipschitz homogeneous with respect to a group. We also prove that certain geodesic spaces…
We prove that the epimorphism relation is a complete analytic quasi-order on the space of countable groups. In the process, we obtain the result of independent interest that the epimorphism relation on pointed reflexive graphs is complete.
We investigate sufficient conditions for the invariance of the real Milnor number under $\mathcal{R}$-bi-Lipschitz equivalence for function-germs $ f, g \colon (\mathbb{R}^n, 0) \to (\mathbb{R}, 0) $. More generally, we explore its…
Let $T$ be an orientation-preserving Lipschitz expanding map of the circle $\T$. A pre-image selector is a map $\tau:\T\to\T$ with finitely many discontinuities, each of which is a jump discontinuity, and such that $\tau(x)\in T^{-1}(x)$…
We introduce the generalized notion of semicontinuity of a function defined on a topological space and derive the useful classification of the so-called Lipschitz derivatives of functions defined on a metric space. Secondly, we investigate…
Contact Geometry is an odd dimensional analogue of Symplectic Geometry. This vague idea can actually be formalized in a rather precise way by means of a Symplectic-to-Contact Dictionary. The aim of this review paper is discussing the basic…
In this paper we provide an up-to-date survey on the study of Lipschitz equivalence of self-similar sets. Lipschitz equivalence is an important property in fractal geometry because it preserves many key properties of fractal sets. A…
We study extension theorems for Lipschitz-type operators acting on metric spaces and with values on spaces of integrable functions. Pointwise domination is not a natural feature of such spaces, and so almost everywhere inequalities and…
In this paper we consider a wide class of generalized Lipschitz extension problems and the corresponding problem of finding absolutely minimal Lipschitz extensions. We prove that if a minimal Lipschitz extension exists, then under certain…