Convex Analysis

Description

The notion of convexity is a notion of great importance in the field of modern analysis. It lies at the cusp of Geometry, Topology and Analysis and at times acts as a bridge between these fields, giving us different viewpoints to consider the same problem.

What is convexity?

One can attribute a notion of convexity to sets and functions:

• A set \(C\) in \(\mathbb{R}^n\) is called a convex set if for any \(\theta \in [0,1]\) and any \(x,y\in C\) we have that $$\theta x + (1-\theta)y \in C.$$ In other words -- if \(x,y\in C\) then the line segment that connects between them lies entirely in \(C\).

  

• A function \(f\) from a convex set \(C\) to \(\mathbb{R}\) is called convex on \(C\) if for any \(\theta \in [0,1]\) and any \(x,y\in C\) we have that $$f\left(\theta x + (1-\theta)y\right) \leq \theta f(x) + (1-\theta)f(y).$$ While this might look daunting, when \(C\) is an interval the above it is nothing more than saying that the straight line connecting the points \(\left(x,f(x) \right)\) and \(\left(y,f(y) \right)\) lies above the graph of the function \(f\) between \(x\) and \(y\).
  

  

Why convexity?

Convex sets and functions enjoy an abundance of useful properties and play an important roles in many topics -- both in theory and applications. For instance:
• Convex sets have many "good" topological properties
• Convex sets and functions are essential in optimisation (such as linear optimisation).
• A local minimum of a convex function is, in fact, a global minimum of the function.
• Convex functions are continuous and almost everywhere differentiable.
and much more.

The goal of this project is to introduce and explore the theory of convex sets and functions.

Group project

The group project will revolve around learning basic concepts and results in the field of convex analysis. By the end of the group project we would have learned
• The definition of a convex function and a convex set.
• Properties of convex functions on \(\mathbb{R}\).
• Affine sets, convex and affine hulls, and cones.
• The epigraph -- the connection between convex functions and convex sets.
• Operations with convex functions.
• Caratheodory's theorem and its consequences.

Mode of Operation and Evidence of Learning for the group project

The project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.

Individual project

The individual project will build on the knowledge we have gained in the group project and will explore additional advanced topics. A few examples of topics you will be able to investigate are:
• Convexity in general normed spaces.
• Separation properties of convex sets (Hahn-Banach Theorem).
• Conjugate functions (Legendre transform) and Fenchel duality.
• Differentiability of convex functions in \(\mathbb{R}^n\).
• Applications to functional inequalities -- Young's inequality, Hölder's inequality, Jensen's inequality, and the Brunn–Minkowski inequality are but a few examples.
Some potential direction would benefit from taking Analysis III together with this project.

Mode of Operation and Evidence of Learning for the individual project

The project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.

Prerequisites and Co-requisites

Prerequisites : Analysis I, Mathematical Methods II and Complex Analysis II.

Additional information

If you would like more information about this project, discuss its scope and/or its prerequisites, don't hesitate to contact me at amit.einav@durham.ac.uk

Resources

• R. T. Rockafellar: Convex Analysis.
• P. M. Gruber, J. M. Wills: Convexity and its applications.
• J. M. Borwein & J. D. Vanderwerff: Convex Functions: Constructions, Characterization and Counterexamples.
• A. V. Arutyunov & V. Obukhovskii: Convex and Set-Valued Analysis.
• B. Simon: Convexity - An Analytic Viewpoint.