Gruber P. Convex and Discrete Geometry

10 Convex Functions
z = f (x) + f ′ − (x)(y − x)
x
z = f (y)
Fig. 1.5. Support and left and right side differentiability
z = f (x) + f ′ + (x)(x − y)
z = f (x) + u(y − x)
It follows from (7) and (8) that a function of the form z = f (x) + u(y − x) for
y ∈ R is an affine support of f at x if and only if f ′ − (x) ≤ u ≤ f ′ + (x). ⊓⊔
As an immediate consequence of this proposition we obtain the following result.
The corresponding d-dimensional result is Theorem 2.7.
Theorem 1.6. Let f : I → R be convex and x ∈ int I . Then the following statements
are equivalent:
(i) f is differentiable at x.
(ii) f has unique affine support at x, say a : R → R, where a(y) = f (x) + u(y − x)
for y ∈ R and u = f ′ (x).
Second-Order Differentiability
We need the following weak notion of twice differentiability: a function f : I → R
is twice (or is second-order) differentiable almost everywhere on I if there are sets
M, N ⊆ I of (Lebesgue) measure 0 such that
f ′ (x) = lim
y→x
and lim
y→x
y∈I \M
f (y) − f (x)
exists for x ∈ I \M,
y − x
f ′ (y) − f ′ (x)
exists for x ∈ I \(M ∪ N).
y − x
The latter limit is denoted by f ′′ (x). (In Sect. 2.2, we will consider a slightly different
notion of twice differentiability almost everywhere for functions of several
variables.)
For the convenience of the reader, we define the Bachmann–Landau symbols o(·)
and O(·):letg, h : I → R and x ∈ I . Then we say that
g(y) = o � h(y) � as y → x, y ∈ I, if |g(y)|
→ 0
|h(y)|
as y → x, y ∈ I, y �= x,
g(y) = O � h(y) � as y → x, y ∈ I if |g(y)| ≤const |h(y)|
for y ∈ I, close to x,
where const is a suitable positive constant.