Sharp estimates for the moduli of continuity of metric projections onto weakly convex sets

G. E. Ivanov

Izvestiya. Mathematics , Tome 79 (2015) no. 4, pp. 668-697.

Résumé

We study the dependence of metric projections on the following three parameters: the point projected, the set to which we are projecting, and the norm (generally speaking, non-symmetric) that determines the metric. We obtain sharp estimates for the moduli of continuity of metric projections onto convex and weakly convex sets in Banach spaces. We also estimate these moduli in terms of the moduli of convexity and smoothness of the space (or the quasi-ball).

Détail

Keywords: metric projection, weakly convex sets.

Bibliographie

[1] Ya. I. Alber, Jen-Chih Yao, “On the projection dynamical systems in Banach spaces”, Taiwanese J. Math., 11:3 (2007), 819–847 | MR | Zbl

[2] B. O. Björnest.{a}l, Continuity of the metric projection operator, preprint, Dept. of Math, TRITA-MAT, 20, Royal Institute of Technology, Stockholm, 1974, 348 pp.

[3] B. O. Björnest.{a}l, “Local Lipschitz continuity of the metric projection operator”, Approximation theory, Papers, VIth semester, Stefan Banach Internat. Math. Center, Warsaw, 1975, Banach Center Publ., 4, PWN, Warsaw, 1979, 43–53 | MR | Zbl

[4] V. I. Berdyshev, “On the modulus of continuity of an operator of best approximation”, Math. Notes, 15:5 (1974), 478–484 | DOI | MR | Zbl

[5] V. I. Berdyshev, “Variation of the norms in the problem of best approximation”, Math. Notes, 29:2 (1981), 95–103 | DOI | MR | Zbl

[6] V. I. Berdyshev, Ustoichivost minimizatsii funktsionalov i ravnomernaya nepreryvnost metricheskoi proektsii, Diss....dokt. fiz.-matem. nauk, Matem. in-t im. V. A. Steklova AN SSSR, M., 1988

[7] Ya. I. Alber, A. I. Notik, “On some estimates for projection operators in Banach space”, Comm. Appl. Nonlinear Anal., 2:1 (1995), 47–55 | MR | Zbl

[8] Ya. I. Alber, “A bound for the modulus of continuity for metric projections in a uniformly convex and uniformly smooth Banach space”, J. Approx. Theory, 85:3 (1996), 237–249 | DOI | MR | Zbl

[9] J.-P. Penot, “Continuity properties of projection operators”, J. Inequal. Appl., 2005:5 (2005), 509–521 | DOI | MR | Zbl

[10] H. Federer, “Curvature measures”, Trans. Amer. Math. Soc., 93:3 (1959), 418–491 | DOI | MR | Zbl

[11] J.-P. Vial, “Strong and weak convexity of sets and functions”, Math. Oper. Res., 8:2 (1983), 231–259 | DOI | MR | Zbl

[12] F. H. Clarke, R. J. Stern, P. R. Wolenski, “Proximal smoothness and lower-$C^{2}$ property”, J. Convex Anal., 2:1-2 (1995), 117–144 | MR | Zbl

[13] F. Bernard, L. Thibault, N. Zlateva, “Characterizations of prox-regular sets in uniformly convex Banach spaces”, J. Convex Anal., 13:3-4 (2006), 525–559 | MR | Zbl

[14] F. Bernard, L. Thibault, N. Zlateva, “Prox-regular sets and epigraphs in uniformly convex Banach spaces: Various regularities and other properties”, Trans. Amer. Math. Soc., 363:4 (2011), 2211–2247 | DOI | MR | Zbl

[15] M. V. Balashov, G. E. Ivanov, “Weakly convex and proximally smooth sets in Banach spaces”, Izv. Math., 73:3 (2009), 455–499 | DOI | DOI | MR | Zbl

[16] V. V. Goncharov, F. F. Pereira, “Neighbourhood retractions of nonconvex sets in a Hilbert space via sublinear functionals”, J. Convex Anal., 18:1 (2011), 1–36 | MR | Zbl

[17] V. V. Goncharov, F. F. Pereira, “Geometric conditions for regularity in a time-minimum problem with constant dynamics”, J. Convex Anal., 19:3 (2012), 631–669 | MR | Zbl

[18] G. Colombo, V. V. Goncharov, B. S. Mordukhovich, “Well-posedness of minimal time problems with constant dynamics in banach spaces”, Set-Valued Var. Anal., 18:3-4 (2010), 349–372 | DOI | MR | Zbl

[19] G. E. Ivanov, M. S. Lopushanski, “O korrektnosti zadach approksimatsii i optimizatsii dlya slabo vypuklykh mnozhestv i funktsii”, Fundament. i prikl. matem., 18:5 (2013), 89–118

[20] G. E. Ivanov, “Weak convexity of sets and functions in a Banach space”, J. Convex Anal., 22:2 (2015), 365–398

[21] T. Abatzoglou, “Continuity of metric projections in uniformly convex and uniformly smooth Banach spaces”, J. Approx. Theory, 39:4 (1983), 299–307 | DOI | MR | Zbl

[22] G. Nordlander, “The modulus of convexity in normed linear spaces”, Ark. Mat., 4:1 (1960), 15–17 | DOI | MR | Zbl

[23] G. E. Ivanov, “On well posed best approximation problems for a nonsymmetric seminorm”, J. Convex Anal., 20:2 (2013), 501–529 | MR | Zbl

[24] B. T. Poljak, “Existence theorems and convergence of minimizing sequences in extremum problems with restrictions”, Soviet Math. Dokl., 7 (1966), 72–75 | MR | Zbl

[25] M. V. Balashov, D. Repovš, “Uniform convexity and the splitting problem for selections”, J. Math. Anal. Appl., 360:1 (2009), 307–316 | DOI | MR | Zbl

[26] O. Hanner, “On the uniform convexity of $L^{p}$ and $l^{p}$”, Ark. Mat., 3:3 (1956), 239–244 | DOI | MR | Zbl

[27] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces, Ergeb. Math. Grenzgeb., II, Function spaces, Springer-Verlag, Berlin–New York, 1979, x+243 pp. | MR | Zbl