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2008•Mathematica BohemicaOpen access

On vector functions of bounded convexity

Libor Veselý, Luděk Zaj́ıček

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 As an application, we give a simple alternative proof of an unpublished result of the first author, containing an estimate of convexity of a composed mapping.

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 As an application, we give a simple alternative proof of an unpublished result of the first author, containing an estimate of convexity of a composed mapping.

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 As an application, we give a simple alternative proof of an unpublished result of the first author, containing an estimate of convexity of a composed mapping.

Key concepts: Convexity, Lipschitz continuity, Bounded function, Mathematics, Normed vector space, Vector-valued function, Convex function, Function (biology)

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