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Proceedings of the American Mathematical Society
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
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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$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0915725-3
PII: S 0002-9939(1988)0915725-3
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