# An almost sharp concentration inequality for random polytopes

and Dirichlet-Voronoi tiling numbers

## Steven Hoehner, Farmingdale State College

## Date and time: 2pm, Friday, February 22, 2019

## Place: CUNY Graduate Center, Rm 4419

Our main contribution is an almost sharp concentration inequality for
the symmetric volume difference of a $ C^2 $ convex body with strictly
positive Gaussian curvature and a circumscribed random polytope with a
restricted number of facets, for any probability measure on the
boundary with a strictly positive density function.

We also show that
the Dirichlet-Voronoi tiling numbers satisfy $ \text{div}_{n-1} =
(2\pi e)^{-1}(n+\ln(n)) + O(1)$, and we provide an interesting
observation and an open conjecture about random partial sphere
"covering" related to results of Erdős, Few and Rogers. This
conjecture is closely connected to the optimality of random polytopes
in high dimensions.

Finally, as an application of all of our results,
we derive a lower bound for the Mahler volume product of polytopes
with a restricted number of vertices.

This is joint work with Gil Kur, MIT (previously at Weizmann Institute of Science).