In his recent book 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
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
q
-positive subset
M
M
in a SSD space
\left( B,\left\lfloor \cdot,\cdot \right\rfloor \right)
(
B
,
⌊
⋅
,
⋅
⌋
)
is the smallest convex representation of
M
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
M
provides a convex representation of
M
M
, too. We also give a new proof of a characterization of maximally
q
q
-positive subsets of SSDB spaces in terms of such special representations