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# Article

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Keywords:
bounded convexity; delta-convex mapping; bounded variation; Banach space
Summary:
Let $X$ be a normed linear space. We investigate properties of vector functions $F\colon [a,b] \to X$ of bounded convexity. In particular, we prove that such functions coincide with the delta-convex mappings admitting a Lipschitz control function, and that convexity $K_a^b F$ is equal to the variation of $F'_+$ on $[a,b)$. As an application, we give a simple alternative proof of an unpublished result of the first author, containing an estimate of convexity of a composed mapping.
References:
[1] Bourbaki, N.: Éléments de Mathématique IX, Livre IV: Fonctions d'une variable réelle (Théorie élémentaire). Second ed., Actualités Scientifiques et Industrielles, vol. 1074, Hermann, Paris (1958).
[2] Chistyakov, V. V.: On mappings of bounded variation. J. Dynam. Control Systems 3 (1997), 261-289. DOI 10.1007/BF02465896 | MR 1449984 | Zbl 0940.26009
[3] Diestel, J., Uhl, Jr., J. J. : The Radon-Nikodým theorem for Banach space valued measures. Rocky Mountain J. Math. 6 (1976), 1-46. DOI 10.1216/RMJ-1976-6-1-1 | MR 0399852 | Zbl 0339.46031
[4] Duda, J.: Curves with finite turn. Czech. Math. J. 58 (2008), 23-49. DOI 10.1007/s10587-008-0003-1 | MR 2402524 | Zbl 1167.46321
[5] Duda, J.: Absolutely continuous functions with values in metric spaces. Real Anal. Exchange 32 (2006-2007), 569-581. MR 2369866
[6] Duda, J., Veselý, L., Zajíček, L.: On d.c. functions and mappings. Atti Sem. Mat. Fis. Univ. Modena 51 (2003), 111-138. MR 1993883 | Zbl 1072.46025
[7] Duda, J., Zajíček, L.: Curves in Banach spaces which allow a $C^2$ parametrization or a parametrization with finite convexity. Preprint (2006), electronically available at {\it http://arxiv.org/abs/math/0603735v1}
[8] Federer, H.: Geometric Measure Theory. Grundlehren der math. Wiss., vol. 153, Springer, New York (1969). MR 0257325 | Zbl 0176.00801
[9] 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
[10] Kirchheim, B.: Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. Amer. Math. Soc. 121 (1994), 113-123. DOI 10.1090/S0002-9939-1994-1189747-7 | MR 1189747 | Zbl 0806.28004
[11] Konyagin, S. V., Veselý, L.: Delta-semidefinite and delta-convex quadratic forms in Banach spaces. Preprint, {\it http://arxiv.org/abs/math/0605549v3} (2007). MR 2398996
[12] Roberts, A. W., Varberg, E. D.: Functions of bounded convexity. Bull. Amer. Math. Soc. 75 (1969), 568-572. DOI 10.1090/S0002-9904-1969-12244-5 | MR 0239021 | Zbl 0176.01204
[13] Roberts, A. W., Varberg, E. D.: Convex Functions. Pure and Applied Mathematics, vol. 57, Academic Press, New York-London (1973). MR 0442824 | Zbl 0271.26009
[14] Veselý, L.: On the multiplicity points of monotone operators on separable Banach spaces. Comment. Math. Univ. Carolin. 27 (1986), 551-570. MR 0873628
[15] Veselý, L.: A short proof of a theorem on compositions of d.c. mappings. Proc. Amer. Math. Soc. 101 (1987), 685-686. DOI 10.2307/2046671 | MR 0911033
[16] Veselý, L.: Topological properties of monotone operators, accretive operators and metric projections. CSc Dissertation (PhD Thesis), Charles University Prague (1990).
[17] Veselý, L., Zajíček, L.: Delta-convex mappings between Banach spaces and applications. Dissertationes Math. (Rozprawy Mat.) 289 (1989), 52 pp. MR 1016045
[18] Veselý, L., Zajíček, L.: On connections between delta-convex mappings and convex operators. Proc. Edinb. Math. Soc. 49 (2006), 739-751. DOI 10.1017/S0013091505000040 | MR 2266160 | Zbl 1115.47048
[19] Veselý, L., Zajíček, L.: On compositions of d.c. functions and mappings. (to appear) in J. Convex Anal. MR 1614031

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