Bénéteau, Laurine Chalopin, Jérémie Chepoi, Victor Vaxès, Yann

The median of a graph $G$ with weighted vertices is the set of all vertices $x$ minimizing the sum of weighted distances from $x$ to the vertices of $G$. For any integer $p\ge 2$, we characterize the graphs in which, with respect to any non-negative weights, median sets always induce connected subgraphs in the $p$th power $G^p$ of $G$. This extends...

Chepoi, Victor Labourel, Arnaud Ratel, Sébastien

$k$-Approximate distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the $k$-approximation of the distance between any two vertices $u$ and $v$ can be determined efficiently by merely inspecting the labels of $u$ and $v$, without using any other information. One of the important problems is f...

Vernicos, Constantin

Alleysson, David

Traduction en français de l'article de Joseph W. Weinberg, The geometry of colors, General Relativity and Gravitation, Vol. 7, No. 1 (1976), pp. 135-169. Avec commentaires et détail des calculs en annexe.

Papadopoulos, Athanase

This is the first of a two-volume set of surveys on geometry in the broad sense (including group actions and topology). The goal is to present a certain number of research topics in a non-technical and appealing manner. The topics surveyed include spherical geometry, the geometry of finite-dimensional normed spaces, metric geometry (Bishop--Gromov ...

Chepoi, Victor Labourel, Arnaud Ratel, Sébastien

Distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the distance between any two vertices $u$ and $v$ can be determined efficiently by merely inspecting the labels of $u$ and $v$, without using any other information. Similarly, routing labeling schemes label the vertices of a graph in a such...

Chalopin, Jérémie Chepoi, Victor Moran, Shay Warmuth, Manfred K.

We examine connections between combinatorial notions that arise in machine learning and topological notions in cubical/simplicial geometry. These connections enable to export results from geometry to machine learning. Our first main result is based on a geometric construction by Tracy Hall (2004) of a partial shelling of the cross-polytope which ca...

Lee, Gye-Seon Marquis, Ludovic Riolo, Stefano

In order to obtain a closed orientable convex projective four-manifold with small positive Euler characteristic, we build an explicit example of convex projective Dehn filling of a cusped hyperbolic four-manifold through a continuous path of projective cone-manifolds.

Gruetzner, Georg

We introduce a notion of discrete-conformal equivalence of closed convex polyhedra in Euclidean 3-space. Using this notion, we prove a uniformization theorem for closed convex polyhedra in Euclidean 3-space.

Steininger, Jakob Yurkevich, Sergey

A polyhedron $\textbf{P} \subset \mathbb{R}^3$ has Rupert's property if a hole can be cut into it, such that a copy of $\textbf{P}$ can pass through this hole. There are several works investigating this property for some specific polyhedra: for example, it is known that all 5 Platonic and 9 out of the 13 Archimedean solids admit Rupert's property. ...