Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
normed space; uniform convexity; closed set; metric projection; $l^p$-space; Fréchet differential; Lipschitz condition
normed space; uniform convexity; closed set; metric projection; $l^p$-space; Fréchet differential; Lipschitz condition
Summary:
Let $F$ be a closed subset of $\mathbb R^n$ and let $P(x) $ denote the metric projection (closest point mapping) of $x\in \mathbb R^n$ onto $F$ in $l^p$-norm. A classical result of Asplund states that $P$ is (Fréchet) differentiable almost everywhere (a.e.) in $\mathbb R^n$ in the Euclidean case $p=2$. We consider the case $2<p<\infty $ and prove that the $i$th component $P_i(x)$ of $P(x)$ is differentiable a.e.\ if $P_i(x)\neq x_i$ and satisfies Hölder condition of order $1/(p-1)$ if $P_i(x)=x_i$.
References:
[1] Abatzoglou, T.: Finite-dimensional Banach spaces with a.e. differentiable metric projection. Proc. Am. Math. Soc. 78 (1980), 492-496. DOI 10.2307/2042418 | MR 0556619 | Zbl 0475.41035
[2] Asplund, E.: Differentiability of the metric projection in finite dimensional metric spaces. Proc. Am. Math. Soc. 38 (1973), 218-219. DOI 10.2307/2038801 | MR 0310150 | Zbl 0269.5200
[3] Clarkson, J.: Uniformly convex spaces. Trans. Am. Math. Soc. 40 (1936), 396-414. DOI 10.2307/1989630 | MR 1501880 | Zbl 0015.35604
[4] Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 153, Springer, Berlin (1969). MR 0257325 | Zbl 0176.00801
[5] Fitzpatrick, S., Phelps, R. R.: Differentiability of metric projection in Hilbert space. Trans. Am. Math. Soc. 270 (1982), 483-501. DOI 10.2307/1999857 | MR 0645326 | Zbl 0504.41029
[6] Hanner, O.: On the uniform convexity of $L^p$ and $l^p$. Ark. Mat. 3 (1956), 239-244. DOI 10.1007/BF02589410 | MR 0077087 | Zbl 0071.32801
[7] Kruskal, J. B.: Two convex counterexamples: A discontinuous envelope function and a non-differentiable nearest point mapping. Proc. Am. Math. Soc. 23 (1969), 697-703. DOI 10.2307/2036613 | MR 0259752 | Zbl 0184.47401
[8] Phelps, R. R.: Convex sets and nearest points. Proc. Am. Math. Soc. 8 (1957), 790-797. DOI 10.2307/2033300 | MR 0087897 | Zbl 0078.35701
[9] Phelps, R. R.: Convex sets and nearest points II. Proc. Am. Math. Soc. 9 (1958), 867-873. DOI 10.2307/2033319 | MR 0104139 | Zbl 0109.14901
[10] Rešetnyak, Ju. G.: Generalized derivatives and differentiability almost everywhere. Mat. Sb., N. Ser. 75(117) (1968), 323-334 Russian. MR 0225159 | Zbl 0165.47202
[11] Shapiro, A.: Differentiability properties of metric projection onto convex sets. J. Optim. Theory Appl. 169 (2016), 953-964. DOI 10.1007/s10957-016-0871-8 | MR 3501393 | Zbl 1342.90192
[12] Zajíček, L.: On differentiation of metric projections in finite dimensional Banach spaces. Czech. Math. J. 33 (1983), 325-336. MR 0718916 | Zbl 0551.41048
Search
Partner of
