Gruber P. Convex and Discrete Geometry

12 Convex Functions
1.3 Convexity Criteria
If a given function is known to be convex, the convexity may yield useful information.
For examples, see the following section. Thus the question arises to find out
whether a function is convex or not. While, in principle, Theorem 1.3 is a convexity
criterion, it is of little practical value.
The property that a convex function has non-decreasing derivative on the set
where the latter exists, and the property that a function which has affine support
everywhere is convex, yield a simple, yet useful convexity criterion which will be
stated below.
Convexity Criteria
Our aim is to prove the following result:
Theorem 1.8. Let I be open and f : I → R differentiable. Then the following
statements are equivalent:
(i) f is convex.
(ii) f ′ is non-decreasing.
Proof. (i)⇒(ii) This follows from Theorem 1.4.
(ii)⇒(i) If (ii) holds, then the first mean value theorem from calculus implies that
for any x ∈ I ,
f (y) = f (x) + f ′� x + ϑ(y − x) � (y − x) ≥ f (x) + f ′ (x)(y − x)
for any y ∈ I and suitable ϑ depending on x and y, where 0