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Journal of Convex Analysis 16 (2009), No. 3, 891--898
Copyright Heldermann Verlag 2009

On Maximally q-Positive Sets

Juan Enrique MartŪnez-Legaz
Dep. d'Economia i d'HistÚria EconÚmica, Universitat AutÚnoma de Barcelona, 08193 Bellaterra, Spain


In his recent book \textit{From Hahn-Banach to monotonicity} (Springer-Verlag, Berlin, 2008), S. Simons has introduced the notion of SSD space to provide an abstract algebraic framework for the study of monotonicity. Graphs of (maximal) monotone operators appear to be (maximally) $q$-positive sets in suitably defined SSD\ spaces. The richer concept of SSDB\ space involves also a Banach space structure. In this paper we prove that the analog of the Fitzpatrick function of a maximally $q$-positive subset $M$ in a SSD space $\left( B,\left\lfloor \cdot ,\cdot \right\rfloor \right)$ is the smallest convex representation of $M$. As a consequence of this result it follows that, in the case of a SSDB space, the conjugate with respect to the pairing $\left \lfloor \cdot, \cdot \right \rfloor$ of any convex representation of $M$ provides a convex representation of $M$, too. We also give a new proof of a characterization of maximally $q$-positive subsets of SSDB spaces in terms of such special representations.

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