On Maximal Domains for C-Convex Functions and Convex Extensions

Abstract

Let

f

be a real valued function with the domain

\dom(f)

in some vector space

X

and let

\mathfrak{C}

be the collection of convex subsets of

X

. The following two questions are investigated; 1. Do there exist maximal convex restrictions

g

of

f

with

\dom(g) \in \mathfrak{C}

? 2. If

f

is convex with

\dom(f)\in \mathfrak{C}

, do there exist maximal convex extension

g

of

f

with

\dom(g)\in \mathfrak{C}

? We will show that the answer to both questions is positive under a certain condition on

\mathfrak{C}

.

Contact details are reproduced from the original publication and may be historical.

School of Mathematics, University of the Witwatersrand, Wits 2050, South Africa

School of Mathematics, University of the Witwatersrand, Wits 2050, South Africa

32E20, 35E10, 26A51, 46A55, 46S40, 52A41

Kh. G. Guseinov, S. A. Duzce, O. Ozer: Convex extensions of the convex set valued maps, J. Math. Anal. Appl. 314 (2006) 672–688.https://doi.org/10.1016/j.jmaa.2005.04.019 Google Scholar
C. Niculescu, L.-E. Persson: Convex Functions and Their Applications, Springer, New York (2006).https://doi.org/10.1007/0-387-31077-0 Google Scholar
H. J. M. Peters, P. P. Wakker: Convex functions on non-convex domains, Econ. Lett. 22 (1987) 251–254.https://doi.org/10.1016/0165-1765(86)90242-9 Google Scholar
W. Rzymowski: Convex extension preserving Lipschitz constant, J. Math. Anal. Appl. 212 (1997) 30–37.https://doi.org/10.1006/jmaa.1997.5093 Google Scholar
M. Tawarmalani, N. V. Sahinidis: Convex extensions and envelopes of lower semicontinuous functions, Math. Program., Ser. A 93 (2002) 247–263.https://doi.org/10.1007/s10107-002-0308-z Google Scholar

Published by Heldermann Verlag, 2010. Rights now held by Banach Press LLC.