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.
As a masters’ student, I got interested this year in convex analysis in relation with FA, precisely. I started to know a bit about the classical results, and the community of researchers that work on this field.
However, I have difficulty assessing whether or not this field is still very active/trendy, and if there are still “basics” open problems to work on – and not very specific problems, yielding huge technicalities, to caricature my question, and motivating applications/connections with other fields. Briefly, whether or not it is a suitable area to start a PhD in ?
(More generally, expect for the trendiest ones, I find it hard to “assess” an area of research, maybe you’d have some hints about it ?)
By the way, thank you for this blog, which I enjoy a lot.
To start a PhD you have to find a potential adviser with whom you have a good relationship. A PhD is a starting point, not an end point. There is also the kind of mathematics. It is considerably easier to find a job in academia with a PhD in statistics than with a PhD is algebraic geometry. Convex analysis is related for instance to statistics and machine learning, a very active field which needs good mathematicians.
Thank you for your answer !
I totally agree with the choice of the advisor being the principal thing, but as it will be next year, I have to target an area first (or at least a few ones), to choose my master’s for next year (after a first one I didn’t like so much), and that’s precisely because I don’t want to end in a algebraic-geometry-like field, that I was asking. As for “usual” convex analysis (convex optimization), it is very clear that it benefits from the huge dynamics of machine learning, I was less sure about the second field (represented by researchers like Klartag or Bobkov, for instance). So it’s very good news if it’s not too hard/blocked.
As a last question (Hope I’m not trying your patience here), do you think an easier way in is through a masters in Analysis, or Probability ? (Of course I guess it depends on the master, but generally)
Probability will give you maybe more possibilities, but actually the quality of teaching and the level is maybe more important than the precise field. Moreover it is important to have a solid background in mathematics, even if you target applied mathematics.
Ok, thanks for sharing your thoughts !