There are two kinds of convex analysis that I know a bit, corresponding to rather distinct scientific communities, taking their roots in the works of Minkowski, Carathéodory, Krein, Fenchel, Tucker, Milman I, and many others. Both domains concern finite dimensional spaces. The first one is more concerned with optimization algorithms in fixed dimension (possibly large nowadays), while the second is related to Banach spaces and involves asymptotic geometric analysis in which the dimension is often high. Both domains are connected to geometry, to probability theory, and to statistics. Here are some of the classical books (not all of them!):
- Convex analysis in relation with optimization.
- Convex analysis, by Rockafellar and Tyrell
- Convex analysis and minimization algorithms I & II, by Hiriart-Urruty and Lemaréchal
- Convex analysis, by Singer
- Convex optimization by Boyd and Vandenberghe
- Variational analysis, by Rockafellar, Tyrrell, and Wets
- Convex analysis in general vector spaces, by Zălinescu
This includes the Hungarian algorithm, the simplex algorithm, the Karush-Kuhn-Tucker (KKT) conditions, the duality in linear programming, and interior point (or barrier) methods. The development of optimization algorithms was considerably accelerated by fast computers with large memory in the past decades (geophysics, computational biology, etc).
- Convex analysis in relation with functional analysis.
- Asymptotic theory of finite-dimensional normed spaces, by Milman II and Schechtman
- The volume of convex bodies and Banach space geometry, by Pisier
- Banach-Mazur distances and finite-dimensional operator ideals, by Tomczak-Jaegermann
- Probability in Banach spaces – Isoperimetry and processes, by Ledoux and Talagrand
- The generic chaining – Upper and lower bounds of stochastic processes, by Talagrand
- The concentration of measure phenomenon, by Ledoux
This includes the Dvoretzky theorem, the Central Limit Theorem for Convex Bodies, the hyperplane conjecture, and the Kannan-Lovász-Simonovits (KLS) conjecture. French readers may take a look at the expository text by Franck Barthe (now a bit obsolet unfortunately). The uniform distribution on a convex body (not only ellipsoids) is a log-concave measure, pulling back the study of such measures to the world of functional analysis.