# Intrinsic and Dual Volume Deviations of Convex Bodies and Polytope

## Steven Hoehner, Longwood University

## Date and time: 2pm, Friday, September 27, 2019

## Place: CUNY Graduate Center, Rm 5383

We establish estimates for the asymptotic best approximation of the
Euclidean unit ball by polytopes under a notion of distance induced by
the intrinsic volumes. We also introduce a notion of distance between
convex bodies that is induced by the Wills functional, and apply it to
derive asymptotically sharp bounds for approximating the ball in high
dimensions. Remarkably, it turns out that there is a polytope which is
almost optimal with respect to all intrinsic volumes simultaneously,
up to an absolute constant.

Finally, we establish asymptotic formulas for the best approximation
of smooth convex bodies by polytopes with respect to a distance
induced by dual volumes, which originate from Lutwak’s dual
Brunn–Minkowski theory.

Joint work with Florian Besau and Gil Kur.