Convex hull deviation and contractibility

Grigory M. Ivanov

DOI: http://dx.doi.org/10.12775/TMNA.2016.089

Abstract


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

Keywords


Hausdorff distance; characterization of inner product spaces; contractibility of a union of balls

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