Abstract
In this paper we initiate a quantitative study of strong proximinality. We define a quantity ϵ(x, t) which we call as modulus of strong proximinality and show that the metric projection onto a strongly proximinal subspace Y of a Banach space X is continuous at x if and only if ϵ(x, t) is continuous at x whenever t > 0. The best possible estimate of ϵ(x, t) characterizes spaces with \(1 \frac{1}{2}\) ball property. Estimates of ϵ(x, t) are obtained for subspaces of uniformly convex spaces and of strongly proximinal subspaces of finite codimension in C(K).
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Dutta, S., Shunmugaraj, P. Modulus of Strong Proximinality and Continuity of Metric Projection. Set-Valued Anal 19, 271–281 (2011). https://doi.org/10.1007/s11228-010-0143-y
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DOI: https://doi.org/10.1007/s11228-010-0143-y