Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
Banach space; Lipschitz surface; d.c. surface; multiplicity points of monotone operators; singular points of convex functions; Aronszajn null sets
Banach space; Lipschitz surface; d.c. surface; multiplicity points of monotone operators; singular points of convex functions; Aronszajn null sets
Summary:
Properties of Lipschitz and d.c. surfaces of finite codimension in a Banach space and properties of generated $\sigma $-ideals are studied. These $\sigma $-ideals naturally appear in the differentiation theory and in the abstract approximation theory. Using these properties, we improve an unpublished result of M. Heisler which gives an alternative proof of a result of D. Preiss on singular points of convex functions.
References:
[1] Berkson, B.: Some metrics on the subspaces of a Banach space. Pacific J. Math. 13 (1963), 7-22. DOI 10.2140/pjm.1963.13.7 | MR 0152869
[2] Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis, Vol. 1. Colloqium publications (American Mathematical Society); v. 48, Providence, Rhode Island (2000). MR 1727673
[3] Duda, J.: On inverses of $\delta$-convex mappings. Comment. Math. Univ. Carolin. 42 (2001), 281-297. MR 1832147 | Zbl 1053.47522
[4] Erdös, P.: On the Hausdorff dimension of some sets in Euclidean space. Bull. Amer. Math. Soc. 52 (1946), 107-109. DOI 10.1090/S0002-9904-1946-08514-6 | MR 0015144
[5] Gohberg, I. C., Krein, M. G.: Fundamental aspects of defect numbers, root numbers, and indexes of linear operators. Uspekhi Mat. Nauk 12 (1957), 43-118 Russian. MR 0096978
[6] Hartman, P.: On functions representable as a difference of convex functions. Pacific J. Math. 9 (1959), 707-713. DOI 10.2140/pjm.1959.9.707 | MR 0110773 | Zbl 0093.06401
[7] Heisler, M.: Some aspects of differentiability in geometry on Banach spaces. Ph.D. thesis, Charles University, Prague (1996).
[8] Kato, T.: Perturbation Theory for Linear Operators. Springer-Verlag, Berin (1976). MR 0407617 | Zbl 0342.47009
[9] Kopecká, E., Malý, J.: Remarks on delta-convex functions. Comment. Math. Univ. Carolin. 31 (1990), 501-510. MR 1078484
[10] Largillier, A.: A note on the gap convergence. Appl. Math. Lett. 7 (1994), 67-71. DOI 10.1016/0893-9659(94)90033-7 | MR 1350148 | Zbl 0804.46026
[11] Lindenstrauss, J., Preiss, D.: Fréchet differentiability of Lipschitz functions (a survey). In: Recent Progress in Functional Analysis, 19-42, North-Holland Math. Stud. 189, North-Holland, Amsterdam (2001). MR 1861745 | Zbl 1037.46043
[12] Lindenstrauss, J., Preiss, D.: On Fréchet differentiability of Lipschitz maps between Banach spaces. Annals Math. 157 (2003), 257-288. DOI 10.4007/annals.2003.157.257 | MR 1954267 | Zbl 1171.46313
[13] Preiss, D.: Almost differentiability of convex functions in Banach spaces and determination of measures by their values on balls. Collection: Geometry of Banach spaces (Strobl, 1989), 237-244, London Math. Soc. Lecture Note Ser. 158 (1990). MR 1110199
[14] Preiss, D., Zajíček, L.: Directional derivatives of Lipschitz functions. Israel J. Math. 125 (2001), 1-27. DOI 10.1007/BF02773371 | MR 1853802
[15] Veselý, L.: On the multiplicity points of monotone operators on separable Banach spaces. Comment. Math. Univ. Carolin. 27 (1986), 551-570. MR 0873628
[16] Veselý, L., Zajíček, L.: Delta-convex mappings between Banach spaces and applications. Dissertationes Math. (Rozprawy Mat.) 289 (1989). MR 1016045
[17] Zajíček, L.: On the points of multivaluedness of metric projections in separable Banach spaces. Comment. Math. Univ. Carolin. 19 (1978), 513-523. MR 0508958
[18] Zajíček, L.: On the points of multiplicity of monotone operators. Comment. Math. Univ. Carolin. 19 (1978), 179-189. MR 0493541
[19] Zajíček, L.: On the differentiation of convex functions in finite and infinite dimensional spaces. Czech. Math. J. 29 (1979), 340-348. MR 0536060
[20] Zajíček, L.: Differentiability of the distance function and points of multi-valuedness of the metric projection in Banach space. Czech. Math. J. 33 (1983), 292-308. MR 0699027
[21] Zajíček, L.: On $\sigma$-porous sets in abstract spaces. Abstract Appl. Analysis 2005 (2005), 509-534. MR 2201041
Search
Partner of
