Uses of convex analysis in Economics
A partial answer: convex analysis is extensively used in axiomatic decision theory, at least in its recent developments. Most of these papers focus on individual behavior. You can have a look for instance at the following papers on ambiguity-averse preferences:


"Maxmin Expected Utility with Non-Unique Prior" (Gilboa &amp; Schmeidler)
"Ambiguity Aversion, Robustness, and the Variational Representation of Preferences" (Maccheroni, Marinacci &amp; Rustichini)
"A Smooth Model of Decision-Making under Ambiguity" (Klibanoff, Marinacci &amp; Mukerji)
"Ambiguity in the Small and in the Large" (Ghirardato and Siniscalchi)


Here is a paper that applies convex analysis to a model of trade under ambiguity aversion: "Subjective Beliefs and Ex-Ante Trade" (Rigotti, Shannon &amp; Strzalecki).

Beyond models of ambiguity aversion, virtually all recent work in axiomatic decision theory makes use of convex analysis, and applies its tools to study various phenomena: regret aversion (Sarver, Ergin), cost of thinking (Ortoleva), random choice (Gul, Pesendorfer)... Please tell me if you want more precise suggestions.

For the mathematical part, a very good reference is Convex Analysis by Rockafellar (1970). It is cited by most of the papers above ;-).