On Maximally q-Positive Sets

J. E. Martínez-Legaz

Journal of convex analysis, Tome 16 (2009) no. 3, pp. 891-898.

Résumé

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.