This is an announcement for the paper "Convex hull deviation and contractibility" by Grigory Ivanov.

Abstract: We study the Hausdorff distance between a set and its convex hull. Let $X$ be a Banach space, define the CHD-module of space $X$ as the supremum of this distance for all subset of the unit ball in $X$. In the case of finite dimensional Banach spaces we obtain the exact upper bound of the CHD-module depending on the dimension of the space. We give an upper bound for the CHD-module in $L_p$ spaces. We prove that CHD-module is not greater than the maximum of the Lipschitz constants of metric projection operator onto hyperplanes. This implies that for a Hilbert space CHD-module equals 1. We prove criterion of the Hilbert space and study the contractibility of proximally smooth sets in uniformly convex and uniformly smooth Banach spaces.

Archive classification: math.FA

Submitted from: grigory.ivanov@phystech.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1501.02596

or

http://arXiv.org/abs/1501.02596