Title: No weak epsilon nets for lines and convex sets in space

Abstract:
One of the most intriguing properties concerning families of convex sets in the d-dimensional space
is the existence of weak ε-nets of constant size. This was discovered by Alon et al. in the early 1990’s.
The weak ε-net theorem plays a fundamental role in modern combinatorial geometry, for instance
in the proof of the celebrated (p, q)-theorem, and determining the correct growth rate of the function
k(ε, d) is recognized as one of the most important open problems in the area.
We prove that there exist no weak ε-nets of constant size for lines and convex sets in the d-dimensional space.

Biography:
Professor at Université de Lorraine, affiliated with École des Mines de Nancy (CS department) and the LORIA laboratory
(joint between Université de Lorraine, CNRS and INRIA).
Formerly, a professor at the computer science department of Université Paris-Est Marne-la-Vallée , now Université Gustave Eiffel (2013–2018).
Co-chair of the program committee of the Symposium of Computational Geometry 2022.
The best paper award at the Symposium of Computational Geometry (SoCG) in 2020, 2018, 2016 and 2012.
Junior member of the Institut Universitaire de France (2014–2019).