  In this paper, we study convex analysis and its theoretical applications. We
first apply important tools of convex analysis to Optimization and to Analysis.
We then show various deep applications of convex analysis and especially
infimal convolution in Monotone Operator Theory. Among other things, we
recapture the Minty surjectivity theorem in Hilbert space, and present a new
proof of the sum theorem in reflexive spaces. More technically, we also discuss
autoconjugate representers for maximally monotone operators. Finally, we
consider various other applications in mathematical analysis.
