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 As an application, we give a simple alternative proof of an unpublished result of the first author, containing an estimate of convexity of a composed mapping.
On vector functions of bounded convexity / L. Vesely, L. Zajicek. - In: MATHEMATICA BOHEMICA. - ISSN 0862-7959. - 133:3(2008), pp. 321-335.
On vector functions of bounded convexity
L. VeselyPrimo
;
2008
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 As an application, we give a simple alternative proof of an unpublished result of the first author, containing an estimate of convexity of a composed mapping.
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