A set is convex if, for any two points within the set, the line segment connecting them lies entirely within the set. This simple yet powerful concept has deep implications in mathematics and other disciplines including economics, computer science, and more.
The goal of the project is to explore fundamental concepts in convex analysis, including
Basic algebraic/topological/extremal properties of convex sets.
Properties of convex functions and duality.
Topics for further investigations could then be chosen according to one's interest. Some possibilities are:
Different aspects of convex geometry (e.g. Brunn-Minkowski theory, isoperimetric problem, Brascamp-Lieb inequalities, Dvoretzky’s theorem).
Differential theory of convex functions and convex optimisation.
Applications in game theory and social science.
Algorithmic aspects of convex geometry (e.g. methods for convex hull).
Stochastic geometry (e.g. integral geometry, convex hull of random points).
Linear Algebra I and Complex Analysis II are essential.
Analysis III is not necessary but strongly recommended. Probability II might be helpful if one is interested in probabilistic questions.
Initial readings could be based on one of the following three books:
Convex Analysis by R. T. Rockafellar.
A classical introduction to convex analysis, with stronger focus on the analysis of convex functions in the second half of the book.
Lectures on Convex Geometry by D. Hug and W. Weil.
An introduction to convex analysis, with stronger emphasis on the geometry of convex sets.
Lectures on Convex Sets by V. Soltan.
A more gentle introduction to the geometry of convex sets.
Further references for applications:
Convex Optimization by S. Boyd and L. Vandenberghe.
A standard textbook/reference for convex optimisation and implementation.
Asymptotic convex geometry: lecture notes by Tomasz Tkocz and Thomas Rothvoss.
Game Theory, Alive by A. R. Karlin and Y. Peres.
Book on various topics in game theory.
Stochastic and Integral Geometry by R. Schneider and W. Weil.
Aspects of Convex Geometry Polyhedra, Linear Programming, Shellings, Voronoi Diagrams, Delaunay Triangulations by J. Gallier and J. Quaintance.