More on the Metric Projection onto a Closed Convex Set in a Hilbert Space

Biagio Ricceri ()
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Biagio Ricceri: University of Catania, Department of Mathematics

A chapter in Contributions in Mathematics and Engineering, 2016, pp 529-534 from Springer

Abstract: Abstract Let H be a real Hilbert space and X a nonempty compact convex subset of H, with 0 ∉ X. For each x ∈ H, denote by P(x) the unique point of X such that $$\|x - P(x)\| = \mbox{ dist}(x,X)$$ . For each r > 0, set $$\gamma (r) =\inf _{\|x\|^{2}=r}\|x - P(x)\|^{2}$$ . Moreover, for each $$\lambda> 1$$ , denote by $$\hat{u}_{\lambda }$$ the unique fixed point of the map $${1 \over \lambda } P$$ . In this paper, in particular, we highlight the following facts: the function $$\lambda \rightarrow h(\lambda ):=\|\hat{ u}_{\lambda }\|^{2}$$ is decreasing in $$]1,+\infty [$$ and its range is $$]0,\|P(0)\|^{2}[$$ ; the function γ is C 1, decreasing and strictly convex in $$]0,\|P(0)\|^{2}[$$ , and one has $$\gamma '(r) = -h^{-1}(r)$$ for all $$r \in ]0,\|P(0)\|^{2}[$$ .

Keywords: Real Hilbert Space; Nonempty Compact Convex Subset; Unique Point; Unique Global Minimum; Wider Literature (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/978-3-319-31317-7_26

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