The study of metric properties of the unit ball (sphere) BV (SV ) of a proper subspace V of a Banach space X has been developed in the last decade. In this paper we give some new results on nearest and farthest points in BV (SV ) to a point x ∈ X\V; in particular we show: a necessary condition for a point to be critical for a distance function, a localization property for nearest and farthest points which leads to a new characterization of Hilbert spaces among Banach or Banach smooth spaces, detailed examples describing the phenomemon of non uniqueness for farthest points.
Farthest and nearest points in balls of subspaces
BARONTI, MARCO;
2015-01-01
Abstract
The study of metric properties of the unit ball (sphere) BV (SV ) of a proper subspace V of a Banach space X has been developed in the last decade. In this paper we give some new results on nearest and farthest points in BV (SV ) to a point x ∈ X\V; in particular we show: a necessary condition for a point to be critical for a distance function, a localization property for nearest and farthest points which leads to a new characterization of Hilbert spaces among Banach or Banach smooth spaces, detailed examples describing the phenomemon of non uniqueness for farthest points.
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