Danylo Radchenko (Université de Lille) : Small volume bodies of constant width

Motivated by applications to Borsuk's conjecture, Schramm has
asked whether a body of constant width 2 in R^n could have exponentially
smaller volume (as function of n) than the ball of radius 1. In this
talk I will give a construction of n-dimensional constant width bodies
whose volume for large n is < 0.891^n times the volume of unit ball,
resolving the question of Schramm.
The talk is based on recent joint work with A. Arman, A. Bondarenko, F.
Nazarov, and A. Prymak.