Journal of Convex Analysis 24 (2017), No. 3, 987--998
Copyright Heldermann Verlag 2017
Convex Compact Sets that Admit a Lower Semicontinuous Strictly Convex Function
Luis Carlos García-Lirola
Dep. de Matemáticas, Facultad de Matemáticas, Universidad de Murcia, 30100 Espinardo-Murcia, Spain
luiscarlos.garcia@um.es
José Orihuela
Dep. de Matemáticas, Facultad de Matemáticas, Universidad de Murcia, 30100 Espinardo-Murcia, Spain
joseori@um.es
Matías Raja
Dep. de Matemáticas, Facultad de Matemáticas, Universidad de Murcia, 30100 Espinardo-Murcia, Spain
matias@um.es
We study the class of compact convex subsets of a topological vector space which admit a strictly convex and lower semicontinuous function. We prove that such a compact set is embeddable in a strictly convex dual Banach space endowed with its weak* topology. In addition, we find many exposed points where a strictly convex lower semicontinuous function is continuous. As a consequence, every set in the above class is the closed convex hull of its exposed points.
Keywords: Convex compact set, convex lower semicontinuous function, exposed point, continuity point.
MSC: 46A55; 46B03,54E35
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