On vector functions of bounded convexity

Libor Veselý; Luděk Zajíček

Mathematica Bohemica (2008)

  • Volume: 133, Issue: 3, page 321-335
  • ISSN: 0862-7959

Abstract

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Let X be a normed linear space. We investigate properties of vector functions F : [ a , b ] → 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 As an application, we give a simple alternative proof of an unpublished result of the first author, containing an estimate of convexity of a composed mapping.

How to cite

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Veselý, Libor, and Zajíček, Luděk. "On vector functions of bounded convexity." Mathematica Bohemica 133.3 (2008): 321-335. <http://eudml.org/doc/250530>.

@article{Veselý2008,
abstract = {Let $X$ be a normed linear space. We investigate properties of vector functions $F\colon [a,b] \rightarrow X$ 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 $F^\{\prime \}_+$ 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 As an application, we give a simple alternative proof of an unpublished result of the first author, containing an estimate of convexity of a composed mapping.},
author = {Veselý, Libor, Zajíček, Luděk},
journal = {Mathematica Bohemica},
keywords = {bounded convexity; delta-convex mapping; bounded variation; Banach space; bounded convexity; delta-convex mapping; bounded variation; Banach space},
language = {eng},
number = {3},
pages = {321-335},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On vector functions of bounded convexity},
url = {http://eudml.org/doc/250530},
volume = {133},
year = {2008},
}

TY - JOUR
AU - Veselý, Libor
AU - Zajíček, Luděk
TI - On vector functions of bounded convexity
JO - Mathematica Bohemica
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 133
IS - 3
SP - 321
EP - 335
AB - Let $X$ be a normed linear space. We investigate properties of vector functions $F\colon [a,b] \rightarrow X$ 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 $F^{\prime }_+$ 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.
LA - eng
KW - bounded convexity; delta-convex mapping; bounded variation; Banach space; bounded convexity; delta-convex mapping; bounded variation; Banach space
UR - http://eudml.org/doc/250530
ER -

References

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  19. Veselý, L., Zajíček, L., On compositions of d.c. functions and mappings, (to appear) in J. Convex Anal. MR1614031

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