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Journal of Convex Analysis 15 (2008), No. 2, 325--343
Copyright Heldermann Verlag 2008

Convex Coradiant Sets with a Continuous Concave Cogauge

Alberto Zaffaroni
Dip. di Scienze Economiche e Matematico-Statistiche, UniversitÓ del Salento, Centro Ecotekne, Via Monteroni, 73100 Lecce, Italy

The paper studies convex coradiant sets and their cogauges. While the concave gauge of a convex coradiant set is superlinear but discontinuous and its Minkowski cogauge is (possibly) continuous but is not concave, we are interested in those convex coradiant sets which admit a continuous concave cogauge. These sets are characterized in primal terms using their outer kernel and in dual terms using their reverse polar set. It is shown that a continuous concave cogauge, if it exists, is not unique; we prove that the class of continuous concave cogauges of some set C admits a greatest element and characterize its support set as the intersection of the reverse polar of C and the polar of its outer kernel.

Keywords: Convex sets, concave gauge, cogauge, radiant sets, coradiant sets, reverse polar.

MSC: 52A07, 46A55, 46B20

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