Gruber P. Convex and Discrete Geometry

2 Convex Functions of Several Variables 23
�
λ µ
λlk(y) + µlk(z) = (λ + µ)lk y +
λ + µ λ + µ z
�
�
λ µ
≤ (λ + µ) f y +
λ + µ λ + µ z
�
�
λ
µ
�
= (λ + µ) f (y − µw) + (z + λw)
λ + µ λ + µ
≤ λf (y − µw) + µf (z + λw)
for y, z ∈ C ∩ Lk and λ, µ > 0 such that y − µw, z + λw ∈ C
by the induction hypothesis and the convexity of f . Hence
lk(y) − f (y − µw) f (z + λw) − lk(z)
≤
µ
λ
for y, z ∈ C ∩ Lk and λ, µ > 0 such that y − µw, z + λw ∈ C.
The supremum of the left-hand side of this inequality is therefore less than or equal
to the infimum of the right-hand side. Thus there is an α ∈ R such that the following
inequalities hold:
lk(y) − f (y − µw)
≤ α for y ∈ C ∩ Lk and µ>0 such that y − µw ∈ C,
µ
f (z + λw) − lk(z)
α ≤ for z ∈ C ∩ Lk and λ>0 such that z + λw ∈ C.
λ
This can also be expressed as follows:
(6) f (z + λw) ≥ lk(z) + αλ
for all z ∈ C ∩ Lk and λ ∈ R such that z + λw ∈ C.
Let Lk+1 be the (k + 1)-dimensional subspace of E d spanned by Lk and w, and let
the linear function lk+1 : Lk+1 → R be defined by
lk+1(z + λw) = lk(z) + αλ for z + λw ∈ Lk+1.
(6) then shows that lk+1 affinely supports f |Lk+1 at o. The induction is complete,
concluding the proof of (5) and thus of the theorem. ⊓⊔
A Characterization of Convex Functions
As a consequence of Theorem 2.3 and the proof of Theorem 1.3, we have the following
equivalence. Clearly, this equivalence can be used to give an alternative definition
of the notion of convex function.
Theorem 2.4. Let C be open and f : C → R. Then the following are equivalent:
(i) f is convex.
(ii) f has affine support at each x ∈ C.