# Gromov's conjecture for uniform contractions revisited

## Károly Bezdek

## Date and time: 2pm, Friday, May 10, 2019

## Place: CUNY Graduate Center, Rm 4419

Gromov's conjecture (1987) states that if the centers of a family of
$N$ congruent balls in $\mathbb{E}^d$ is contracted, then the volume
of the intersection does not decrease. A uniform contraction is a
contraction where all the pairwise distances in the first set of
centers are larger than all the pairwise distances in the second set
of centers, that is, when the pairwise distances of the two sets are
separated by some positive real number. The speaker and M. Naszódi
[Discrete Comput. Geom. 60/4 (2018), 967-980] proved Gromov's
conjecture for all uniform contractions in $\mathbb{E}^d$, $d>1$ under
the condition that $ N\geq \left(1+\sqrt{2}\right)^d$. In this talk, I
will improve this result in dimensions 2 and 3 as well as in high
dimension.