### About this book

This book deals with the study of convex functions and of their
behavior from the point of view of stability with respect to
perturbations. Convex functions are considered from the modern point
of view that underlines the geometrical aspect: thus a function is
defined as convex whenever its graph is a convex set.

A primary goal of this book is to study the problems of stability and
well-posedness, in the convex case. Stability means that the basic
parameters of a minimum problem do not vary much if we slightly change
the initial data. On the other hand, well-posedness means that points
with values close to the value of the problem must be close to actual
solutions. In studying this, one is naturally led to consider
perturbations of functions and of sets. This approach fits perfectly
with the idea of regarding functions as sets. Thus the second part of
the book starts with a short, yet rather complete, overview of the
so-called hypertopologies, i.e. topologies in the closed subsets of a
metric space.

While there exist numerous classic texts on the issue of stability,
there only exists one book on hypertopologies [Beer 1993]. The current
book differs from Beer’s in that it contains a much more condensed
explication of hypertopologies and is intended to help those not
familiar with hypertopologies learn how to use them in the context of
optimization problems.

Written for:

Graduate students, mathematicians