Periodic cellular materials allow triggering complex elastic behaviors within the volume of a part. In this work, we study a novel type of 3D periodic cellular materials that emerge from a growth process in a lattice. The growth is parameterized by a 3D star-shaped set at each lattice point, defining the geometry that will appear around it. Individ...
Published in Journal of Fixed Point Theory and Applications
This note discusses some aspects of the asymptotic behaviour of nonexpansive maps. Using metric functionals, we make a connection to the invariant subspace problem and prove a new result for nonexpansive maps of ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \...
Starting with a lattice with an action of $\mathbb{Z}$ or $\mathbb{R}$, we build a Helly graph or an injective metric space. We deduce that the $\ell^\infty$ orthoscheme complex of any bounded graded lattice is injective. We also prove a Cartan-Hadamard result for locally injective metric spaces. We apply this to show that any Garside group acts on...
Local symmetries are primarily defined in the case of spacetime, but several authors have defined them outside this context, sometimes with the help of groupoids. We show that, in many cases, local symmetries can be defined as global symmetries. We also show that groups can be used, rather than groupoids, to handle local symmetries. Examples are gi...
The purpose of this report is to acknowledge the influence of M. Gromov's vision of geometry on our own works. It is two-fold: in the first part we aim at describing some results, in dimension 3, around the question: which open 3-manifolds carry a complete Riemannian metric of positive or non negative scalar curvature? In the second part we look fo...
This article introduces the notion of L-tangle-free compact hyperbolic surfaces, inspired by the identically named property for regular graphs. Random surfaces of genus g, picked with the Weil-Petersson probability measure, are (a log g)-tangle-free for any a
The Gromov-Hausdorff space is usually defined in textbooks as "the space of all compact metric spaces up to isometry". We describe a formalization of this notion in the Lean proof assistant, insisting on how we need to depart from the usual informal viewpoint of mathematicians on this object to get a rigorous formalization.
Topological data analysis (or TDA for short) consists in a set of methods aiming to extract topological and geometric information from complex nonlinear datasets. This field is here tackled from two different perspectives.First, we consider techniques from geometric inference, whose goal is to reconstruct geometric invariants of a manifold thanks t...
A masure (a.k.a affine ordered hovel) I is a generalization of the Bruhat-Tits building that is associated to a split Kac-Moody group G over a non-archimedean local field. This is a union of affine spaces called apartments. When G is a reductive group, I is a building and there is a G-invariant distance inducing a norm on each apartment. In this pa...
Pick $n$ points $Z_0,...,Z_{n-1}$ uniformly and independently at random in a compact convex set $H$ with non empty interior of the plane, and let $Q^n_H$ be the probability that the $Z_i$'s are the vertices of a convex polygon. Blaschke 1917 \cite{Bla} proved that $Q^4_T\leq Q^4_H\leq Q^4_D$, where $D$ is a disk and $T$ a triangle. In the present p...
