Convex hull deviation and contractibility

TL;DR

This paper investigates the Hausdorff distance between sets and their convex hulls in Banach spaces, providing bounds, characterizations, and criteria related to convexity, contractibility, and geometric properties.

Contribution

It introduces the CHD-module, establishes exact bounds in finite dimensions, and explores contractibility criteria in various Banach space settings.

Findings

Exact upper bounds for CHD-module in finite-dimensional Banach spaces

Upper bounds for CHD-module in L_p spaces

CHD-module equals 1 in Hilbert spaces

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.