Gruber P. Convex and Discrete Geometry

2 Convex Functions of Several Variables 35
(ii) F separates points, i.e. for any x, y ∈ S, x �= y, there is a function
f ∈ F with f (x) �= f (y).
(iii) F vanishes nowhere on S, i.e. for any x ∈ S there is a function
f ∈ F with f (x) �= 0.
Then F is dense in the space of all real continuous functions on S, endowed
with the maximum norm.
Next, some simple properties of convex functions will be given:
(2) Let f : C → R be convex and f ≥ 0. Then f 2 is also convex.
The convexity and the non-negativity of f together with the fact that the function
t → t 2 for t ≥ 0 is non-decreasing and convex, imply (2):
f � (1 − λ)x + λy � 2 ≤ � (1 − λ) f (x) + λf (y) � 2 ≤ (1 − λ) f (x) 2 + λf (y) 2
for x, y ∈ C, 0 ≤ λ ≤ 1.
The following proposition is an immediate consequence of (2), the fact that sums of
convex functions are again convex, and of the simple identity fg = 1 �
2 ( f + g) 2 −
( f 2 + g2 ) � :
(3) Let f, g : C → R be convex and f, g ≥ 0. Then fg ∈ D.
In the last part of the proof it will be shown that D has Properties (i)–(iii) in
Stone’s theorem (1).
(4) D has Property (i).
Only multiplication has to be justified. Let f − g, h − k ∈ D. The convex functions
f, g, h, k : C → R are continuous on the compact set C by assumption and thus
are bounded. After adding the same suitable constant to each of these functions and
changing notation if this constant is �= 0, we may assume that f, g, h, k ≥ 0. Then
( f − g)(h − k) = fh+ gk − fk− gh ∈ D by (3), concluding the proof of (4).
(5) D has Properties (ii) and (iii).
This follows by considering, for example, affine functions on C, which are clearly
convex.
Having proved (4) and (5), the theorem is an immediate consequence of Stone’s
theorem (1). ⊓⊔
Remark. The above proof shows that Theorem 2.11 actually holds in spaces which
are more general than E d .
2.5 A Sufficient Condition of Courant and Hilbert in the Calculus
of Variations
For many extremum problems necessary conditions for solutions are well known and
easy to obtain. In general, it is more difficult to give sufficient conditions. Consider
the following simple example: Let f : I → R be differentiable. Then, if f attains a