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On the almost everywhere differentiability of the metric projection on closed sets in lp(ℝn), 2 < p < ∞
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2018 (English)In: Czechoslovak Mathematical Journal, ISSN 0011-4642, E-ISSN 1572-9141, Vol. 68, no 143, p. 943-951Article in journal (Refereed) Published
Abstract [en]

Let F be a closed subset of ℝn and let P(x) denote the metric projection (closest point mapping) of x ∈ ℝn onto F in lp-norm. A classical result of Asplund states that P is (Fréchet) differentiable almost everywhere (a.e.) in ℝn in the Euclidean case p = 2. We consider the case 2 < p < ∞ and prove that the ith component Pi(x) of P(x) is differentiable a.e. if Pi(x) 6= xi and satisfies Hölder condition of order 1/(p−1) if Pi(x) = xi.

Place, publisher, year, edition, pages
Berlin/Heidelberg: Springer, 2018. Vol. 68, no 143, p. 943-951
Keywords [en]
normed space, uniform convexity, closed set, metric projection, l(p)-space, Frechet differential, Lipschitz condition
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-153050DOI: 10.21136/CMJ.2018.0038-17ISI: 000451778500004OAI: oai:DiVA.org:umu-153050DiVA, id: diva2:1260615
Available from: 2018-11-04 Created: 2018-11-04 Last updated: 2018-12-18Bibliographically approved

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Sjödin, Tord

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