Abstract
The study of metric properties of the unit ball (sphere) \(B_V (S_V)\) 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 \(B_V (S_V)\) to a point \(x \in X{\setminus } V\); 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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