Convex hull deviation and contractibility
Słowa kluczowe
Hausdorff distance, characterization of inner product spaces, contractibility of a union of ballsAbstrakt
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.Pobrania
Opublikowane
2017-07-07
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1.
IVANOV, Grigory M. Convex hull deviation and contractibility. Topological Methods in Nonlinear Analysis [online]. 7 lipiec 2017, T. 50, nr 1, s. 9–25. [udostępniono 22.7.2024].
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