Gruber P. Convex and Discrete Geometry

A Characterization of Convex Functions
1 Convex Functions of One Variable 7
The concept of affine support can be used to define the convexity of a function, as
the next result shows. Theorem 2.4 is the corresponding d-dimensional result and the
corresponding result for convex bodies is Theorem 4.2.
Theorem 1.3. Let I be open and f : I → R. Then the following statements are
equivalent:
(i) f is convex.
(ii) f has affine support at each x ∈ I.
Proof. (i) ⇒ (ii) This follows from Theorem 1.2.
(ii) ⇒ (i) If f has affine support at each x ∈ I ,sayax(·), then clearly,
f (y) = sup{ax(y) : x ∈ I } for y ∈ I.
As the supremum of a family of affine and thus convex functions, f is also convex:
f � (1 − λ)y + λz � = sup � � � �
ax (1 − λ)y + λz : x ∈ I
= sup � (1 − λ)ax(y) + λax(z) : x ∈ I �
≤ (1 − λ) sup � ax(y) : x ∈ I }+λ sup{ax(z) : x ∈ I �
= (1 − λ) f (y) + λf (z) for y, z ∈ I, 0 ≤ λ ≤ 1. ⊓⊔
It is sufficient in Theorem 1.3 to assume that f is affinely supported locally at
each point of I .
First-Order Differentiability
In the following several well-known results from analysis will be used. A reference
for these is [499].
A theorem of Lebesgue says that an absolutely continuous real function on an
interval is almost everywhere differentiable. This combined with Theorem 1.1 shows
that a convex function f : I → R is almost everywhere differentiable. Yet, as we
shall see below, the convexity of f yields an even stronger result: f is differentiable
at each point of I with, at most, a countable set of exceptions.
In order to state this result in a precise form we need the notions of left and right
derivative f ′ − (x) and f ′ + (x) of a function f : I → R at a point x ∈ I :
f ′ − (x) = lim
y→x−0
f (y) − f (x)
y − x
(x not the left endpoint of I ),
f ′ + (x) = lim
y→x+0
f (y) − f (x)
y − x
(x not the right endpoint of I ).
The left and right derivatives may or may not exist.