Convex hull deviation and contractibility
Keywords
Hausdorff distance, characterization of inner product spaces, contractibility of a union of ballsAbstract
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.Published
2017-07-07
How to Cite
1.
IVANOV, Grigory M. Convex hull deviation and contractibility. Topological Methods in Nonlinear Analysis. Online. 7 July 2017. Vol. 50, no. 1, pp. 9 - 25. [Accessed 28 March 2024].
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