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On almost everywhere differentiability of the metric projection on closed sets in lp(ℝn), 2 < p < ∞

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Abstract

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.

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  1. Department of Mathematics, Umeå University, Umeå, 901 87, Sweden

    Tord Sjödin

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  1. Tord Sjödin
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Correspondence to Tord Sjödin.

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Sjödin, T. On almost everywhere differentiability of the metric projection on closed sets in lp(ℝn), 2 < p < ∞. Czech Math J 68, 943–951 (2018). https://doi.org/10.21136/CMJ.2018.0038-17

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  • DOI: https://doi.org/10.21136/CMJ.2018.0038-17

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