Gruber P. Convex and Discrete Geometry

6 Convex Functions
Together these inequalities imply that
| f (x) − f (y)|
|x − y|
��
�� f (v) − f (u)
� �
� �
≤ max
�, �
v − u
f (z) − f (w)
z − w
��
�
� = L, say. ⊓⊔
If a is an endpoint of I , a convex function f : I → R may not be continuous
at a. A simple example is provided by the function f : [0, 1] → R defined by
f (0) = 1, f (x) = 0forx ∈ (0, 1].
Support Properties
A function f : I → R has affine support at a point x ∈ I if there is an affine function
a : R → R of the form a(y) = f (x) + u(y − x) for y ∈ R where u is a suitable
constant, such that
f (y) ≥ a(y) = f (x) + u(y − x) for y ∈ I.
The affine function a is called an affine support of f at x. The geometric notion of
affine support is intimately connected with the notion of convexity.
Our first result shows that a convex function has affine support at each point in the
interior of its interval of definition. This result may be considered as a 1-dimensional
Hahn–Banach theorem. The corresponding d-dimensional result is Theorem 2.3 and
Theorem 4.1 is the corresponding result for convex bodies.
Theorem 1.2. Let f : I → R be convex and x ∈ int I . Then f has affine support
at x.
Proof. We may suppose that x = 0 and f (0) = 0. Let w ∈ I,w �= 0. Then the
convexity of f implies that
or
�
λ
µ
0 = (λ + µ) f (−µw) +
λ + µ λ + µ (λw)
�
≤ λf (−µw) + µf (λw) for λ, µ > 0, where λw, −µw ∈ I,
− f (−µw)
µ
≤
f (λw)
λ
for λ, µ > 0, where λw, −µw ∈ I.
The supremum (over µ) of the left-hand side is therefore less than or equal to the
infimum (over λ) of the right-hand side. Hence we may choose α ∈ R such that
Equivalently,
− f (−µw)
µ
≤ α ≤
f (λw)
λ
for λ, µ > 0, where λw, −µw ∈ I.
f (λw) ≥ αλ for λ ∈ R , where λw ∈ I.
Thus a(λw) = αλ for λ ∈ R is an affine support of f at x = 0. ⊓⊔