L. Pournin

activity

20122026

most citedThe diameter of associahedra

118 citations · 220 across the 30 of their papers we have counts for

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12 papers · 1 filter

math.MG2026

Flat simplices and kissing polytopes

Antoine Deza, Lionel Pournin

We consider how flat a lattice simplex contained in the hypercube $[0, k]^{d}$ can be. This question is related to the notion of kissing polytopes: two lattice polytopes contained in…

math.MG2025

Kissing polytopes in dimension 3

Antoine Deza, Zhongyuan Liu, Lionel Pournin

It is shown that the smallest possible distance between two disjoint lattice polytopes contained in the cube $[0, k]^{3}$ is exactly $\frac{1}{2 ( 2 k ^{2} - 4 k + 5 ) ( 2 k ^{2} - 2 k + 1 )}$ for…

math.MG2024

Deep sections of the hypercube

Lionel Pournin

Consider a non-negative number $t$ and a hyperplane $H$ of $R^{d}$ whose distance to the center of the hypercube $[0, 1]^{d}$ is $t$ . If $t$ is equal to $0$ and $H$ is orthogon…

math.MG2023

Many equiprojective polytopes

Théophile Buffière, Lionel Pournin

A $3$ -dimensional polytope $P$ is $k$ -equiprojective when the projection of $P$ along any line that is not parallel to a facet of $P$ is a polygon with $k$ vertices. In 1968, Geoff…

math.MG2023★ 2 cited

Kissing polytopes

Antoine Deza, Shmuel Onn, Sebastian Pokutta +1

We investigate the following question: how close can two disjoint lattice polytopes contained in a fixed hypercube be? This question stems from various contexts where the minimal d…

math.MG2022★ 6 cited

Local extrema for hypercube sections

Lionel Pournin

Consider the hyperplanes at a fixed distance $t$ from the center of the hypercube $[0, 1]^{d}$ . Significant attention has been given to determining the hyperplanes $H$ among these suc…

math.MG2021★ 8 cited

Shallow sections of the hypercube

Lionel Pournin

Consider a $d$ -dimensional closed ball $B$ whose center coincides with that of the hypercube $[0, 1]^{d}$ . Pick the radius of $B$ in such a way that the vertices of the hypercube are…

math.MG2020

Polytopal balls arising in optimization

Antoine Deza, Jean-Baptiste Hiriart-Urruty, Lionel Pournin

We study a family of polytopes and their duals, that appear in various optimization problems as the unit balls for certain norms. These two families interpolate between the hypercu…

math.MG2018★ 4 cited

Elementary moves on lattice polytopes

Julien David, Lionel Pournin, Rado Rakotonarivo

We introduce a graph structure on Euclidean polytopes. The vertices of this graph are the $d$ -dimensional polytopes contained in $R^{d}$ and its edges connect any two polyto…

math.MG2018★ 3 cited

Diameter, decomposability, and Minkowski sums of polytopes

Antoine Deza, Lionel Pournin

We investigate how the Minkowski sum of two polytopes affects their graph and, in particular, their diameter. We show that the diameter of the Minkowski sum is bounded below by the…

math.MG2016★ 14 cited

Improved bounds on the diameter of lattice polytopes

Antoine Deza, Lionel Pournin

We show that the largest possible diameter $δ (d, k)$ of a $d$ -dimensional polytope whose vertices have integer coordinates ranging between $0$ and $k$ is at most $kd-\lceil2d/3\rcei…

math.MG2012★ 3 cited

The flip-graph of the 4-dimensional cube is connected

Lionel Pournin

Flip-graph connectedness is established here for the vertex set of the 4-dimensional cube. It is found as a consequence that this vertex set has 92 487 256 triangulations, partitio…