Gruber P. Convex and Discrete Geometry

Convex Functions
Convex functions came to the face rather late. Early contributions are due to Stolz
[972], Brunn [174], Hadamard [459] and Jensen [544] around the beginning of the
twentieth century. The systematic study of convex functions dealing with continuity
and differentiability properties, with variants of the fundamental notions and with
inequalities began only in the twentieth century. The unpublished seminal lecture
notes of Fenchel [334] in the early 1950s led to convex analysis, a careful study of
analytic properties of convex functions related to optimization. Major contributions
were made by Moreau, Rockafellar and Phelps.
Convex functions appear and are useful in many areas of mathematics, including
the calculus of variations, control theory, inequalities and functional equations, optimization
and econometrics. On the other hand, investigation of convex functions per
se has led to a rich and voluminous theory.
This chapter contains basic geometric and analytic properties of convex functions
of one and of several variables, related to convex geometry. A highlight
is Alexandrov’s theorem on second-order differentiability almost everywhere of a
convex function. A further result is of Stone–Weierstrass type. Applications treat
inequalities, the characterization of the gamma function by Bohr and Mollerup, and
a sufficient condition in the calculus of variations due to Courant and Hilbert.
For more detailed expositions and references on convex functions with an emphasis
on convexity, consult the articles and books of Beckenbach [87], Roberts
and Varberg [841], Roberts [840] and also Giles [379], Van Tiel [1006] and
Czerwik [234].
We will not treat convex functions in the sense of convex analysis and general
analysis. For material in this direction, see Moreau [755], Rockafellar [843] and
Phelps [800], Giles [379], Van Tiel [1006], Borwein and Lewis [158], Hiriart-Urruty
and Lemaréchal [505], Magaril-Il’yaev and Tikhomirov [678] and Niculescu and
Persson [770].
Throughout the book, vectors are columns, unless stated otherwise, but for convenience,
we write vectors as rows as usual. If there is need to stress, that a row
actually means a column, the superscript T is added.