### Convex hull deviation and contractibility

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.

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

### Refbacks

- There are currently no refbacks.