Convexity
Convexity is a beautifiul and useful subject at the intersection of geometry and analysis. Nominally, convex geometry studies convex sets and convex analysis studies convex functions, but these two notions are closed connected. As often the case, the geometry is most helpful for understanding and the analysis most helpful for calculating.
TODO: Dictionary between sets and functions
Convex analysis
Many works on convex optimization are also relevant.
Surveys
- Hiriart-Urruty 2013: Convex analysis and optimization in the past 50 years
(doi)
- A brief personal history of convex analysis starting from the early 60s
Books
- Rockafellar, 1970: Convex analysis
- A famous book, long the standard reference
- Rockafellar & Wets, 1998: Variational analysis (doi)
- Hiriart-Urruty & Lemarechal, 1993: Convex analysis and minimization
algorithms, I (doi) and II (doi)
- Abridged in single volume: Hiriart-Urruty & Lemarechal, 2001: Fundamentals of convex analysis (doi)
- My preferred reference, with a nice of mix of the geometric and analytic
- Simon, 2011: Convexity: An analytic viewpoint (doi)
Convex geometry
Much modern literature is about high-dimensional convex geometry:
- Ball, 1997: An elementary introduction to modern convex geometry (pdf)
- Chapter in MSRI’s lecture notes , Flavors of geometry
- Barvinok, 2002: A course in convexity (doi)