Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the ARG MIN multifunction for lower semicontinuous functions

Authors: Gerald Beer and Petar Kenderov
Journal: Proc. Amer. Math. Soc. 102 (1988), 107-113
MSC: Primary 26A15,; Secondary 49A50,54C60,90C48
MathSciNet review: 915725
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The epi-topology on the lower semicontinuous functions $ L\left( X \right)$ on a Hausdorff space $ X$ is the restriction of the Fell topology on the closed subsets of $ X \times R$ to $ L\left( X \right)$, identifying lower semicontinuous functions with their epigraphs. For each $ f \in L\left( X \right)$, let arg min $ f$ be the set of minimizers of $ f$. With respect to the epi-topology, the graph of arg min is a closed subset of $ L\left( X \right) \times X$ if and only if $ X$ is locally compact. Moreover, if $ X$ is locally compact, then the epi-topology is the weakest topology on $ L\left( X \right)$ for which the arg min multifunction has closed graph, and the operators $ f \to f \vee g$ and $ f \to f \wedge g$ are continuous for each continuous real function $ g$ on $ X$.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26A15,, 49A50,54C60,90C48

Retrieve articles in all journals with MSC: 26A15,, 49A50,54C60,90C48

Additional Information

Keywords: arg min multifunction, lower semicontinuous function, epiconvergence, epi-topology, Fell topology, function lattice, locally compact Hausdorff space
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society