MATHEMATICA BOHEMICA, Vol. 133, No. 3, pp. 321-335 (2008)

On vector functions of bounded convexity

Libor Vesely, Ludek Zajicek

Libor Vesely, Dipartimento di Matematica, Universita degli Studi di Milano, Via C. Saldini 50, 201 33 Milano, Italy, e-mail:, Ludek Zajicek, Charles University, Faculty of Mathematics and Physics, Sokolovska 83, 186 75 Praha 8, Czech Republic, e-mail:

Abstract: 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 As an application, we give a simple alternative proof of an unpublished result of the first author, containing an estimate of convexity of a composed mapping.

Keywords: bounded convexity, delta-convex mapping, bounded variation, Banach space

Classification (MSC2000): 47H99, 26A99

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