Gruber P. Convex and Discrete Geometry

22 Convex Functions
| f (z) − f (y)| ≤L�z − y� where L = 2γ
ε .
This concludes the proof of (4) and thus of (1). ⊓⊔
Remark. Note that continuity may not extend to bd C as the following example
shows: let B d be the solid Euclidean unit ball in E d and f : B d → R. Suppose that
for a suitable real α>0 the restriction of f to int B d is convex and bounded above by
α and the restriction of f to bd B d is bounded below by 2α, but otherwise arbitrary.
Then f is convex and discontinuous at each point of bd B d .
Support and a Hahn–Banach Type Result
Support and the related separation properties of convex functions and convex sets
are of great importance for convex analysis. Moreover, they are bridges between
convex geometry and optimization and – to some extent – functional analysis. See,
e.g. Hiriart-Urruty and Lemaréchal [505], Stoer and Witzgall [970] and Rudin [861].
A function f : C → R has affine support at a point x ∈ C if there is an affine
function a : E d → R of the form a(y) = f (x) + u · (y − x) for y ∈ E d , where u is
a suitable vector in E d , such that
f (y) ≥ a(y) = f (x) + u · (y − x) for y ∈ C.
a is called an affine support of f at x. The dot · denotes the usual inner product in
E d .
The following result extends Theorem 1.2. It is a finite-dimensional version of
the Hahn–Banach theorem. For the general Hahn–Banach theorem see, e.g. [861].
Our proof is by induction. It generalizes to the infinite dimensional case where, of
course, induction has to be replaced by transfinite induction. A related result on the
existence of support hyperplanes of a convex body is Theorem 4.1. Our proof of
the latter result is essentially finite dimensional and thus basically different from the
proof of the following result.
Theorem 2.3. Let f : C → R be convex and P an affine subspace in E d through a
point x ∈ int C. Suppose that the restriction f |P has an affine support aP at x. Then
f has an affine support a at x which extends aP, i.e. a|P = aP.
Proof. We may assume that x = o and f (o) = 0. Let L = P. Let dim stand
for dimension. If dim L = d, we are done. Otherwise it is sufficient to prove the
following proposition:
(5) Let k = dim L, dim L + 1,...,d. Then there are a k-dimensional linear
subspace Lk ⊇ L and a linear function lk : Lk → R which affinely supports
f |Lk and extends lL = aP.
The proof of (5) is by induction. If k = dim L, (5) holds by assumption. Suppose
now that (5) holds for a k < d. Choose w ∈ C\Lk. Then