Raj, Shivam and Paul, Tanmoy
(2018)
Analysis of Convex Functions.
Masters thesis, Indian Institute of Technology Hyderabad.
Abstract
Convexity is an old subject in mathematics. The �rst speci�c de�nition of convexity
was given by Herman Minkowski in 1896. Convex functions were introduced by Jensen
in 1905. The concept appeared intermittently through the centuries, but the subject
was not really formalized until the seminal 1934 tract Theorie der konverxen Korper of
Bonneson and Fenchel.Today convex geometry is a mathematical subject in its own right. Classically oriented
treatments, like the work done by Frederick Valentine form the elementary de�nition,
which is that a domain K in the plane or in RN is convex if for all P;Q 2 K, then the
segment PQ connecting P to Q also lies in K. In fact this very simple idea gives forth
a very rich theory. But it is not a theory that interacts naturally with mathematical
analysis. For analysis, one would like a way to think about convexity that is expressed
in the language of functions and perhaps its derivatives.
Our goal in this thesis is to present and to study convexity in a more analytic way.
Through Chapter 1, Chapter 2 and Chapter 3, I have tried to point out the important role
of convex sets and its associated convex functions in Mathematical Analysis. Chapter 1 is
devoted to Convex sets and some geometric properties achieved by these objects in �nite
Euclidean spaces. The emphasis is given on establishing a criteria for convexity. Various
useful examples are given, and it is shown how further examples can be generated from
these by means of operations such as addition or taking convex hulls. The fundamental
idea to be understood is that the convex functions on Rn can be identi�ed with certain
convex subsets of Rn+1 (their epigraphs), while the convex sets in Rn can be identi�ed
with certain convex functions on Rn (their indicators). These identi�cations make it easy
to pass back and forth between a geometric approach and an analytic approach. Chapter 2
begins with idea of convexity of functions in a �nite dimensional space. Convex functions
are an important device for the study of extremal problems. They are also important
analytic tools. The fact that a convex function can have at most one minimum and
no maxima is a notable piece of information that proves to be quite useful. A convex
function is also characterized by the non negativity of its second derivative. This useful
information interacts nicely with the ideas of calculus. We relate convex functions to
an elegant characterisation of Gamma functions by Bohr Mollerup Theorem. Chapter 3
provides an introduction to convex analysis, the properties of sets and functions in in�nite
dimensional space. We start by taking the convexity of the epigraph to be the definition
of a convex function, and allow convex functions to be extended -real valued. One of the
main themes of this chapter is the maximization of linear functions over non empty convex
sets. Here we relate the subdi�erential to the directional derivative of a function. There
are several modern works on convexity that arise from studies of functional analysis.
One of the nice features of the analytic way of looking at convexity is the Bishop-Phelps
Theorem, it says that in a Banach Space, a convex function has a subgradient on a dense
subset of its e�ective.
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