# Gromov's conjecture for uniform contractions revisited

## 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.