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Convex analysis

Dodecahedron

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!):

5 Comments

  1. Gericault 2018-05-03

    Hello,
    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.

  2. Djalil Chafaï 2018-05-04

    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.

  3. Gericault 2018-05-05

    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)

  4. Djalil Chafaï 2018-05-05

    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.

  5. Gericault 2018-05-05

    Ok, thanks for sharing your thoughts !

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