Gruber P. Convex and Discrete Geometry

34 Convex Functions
2.4 A Stone–Weierstrass Type Theorem
The problem, whether interesting special families of functions are dense in the class
of real or complex continuous functions on a given space, has drawn attention ever
since Weierstrass gave his well known approximation theorem.
In this section, we present a Stone-Weierstrass type result which shows that
differences of convex functions are dense in the space of continuous functions.
A Stone–Weierstrass Type Theorem for Convex Functions
For the following result, see Alfsen [21].
Theorem 2.11. Let C be compact and convex. Then the set D of all differences
of continuous convex functions on C is dense in the space of all real continuous
functions on C, endowed with the maximum norm.
We present two proofs. The first one was proposed by Schneider [910] and makes
use of the Weierstrass approximation theorem for several variables and the convexity
criterion of Brunn and Hadamard. The second proof is based on a theorem of Stone
of Stone–Weierstrass type.
Proof (using the Weierstrass approximation theorem). We need the following version
of the approximation theorem:
The family of all real polynomials in d variables on the cube K ={x :
|xi| ≤1} is dense in the space of all real continuous functions on K .
For the proof of the theorem we may assume that C ⊆ K . Since each continuous real
function on the compact set C can be continuously extended to K ,itfollowsthat
The family of all (restrictions of) real polynomials on C is dense in the
space of all real continuous functions on C.
This yields the theorem if we can show that
Each polynomial p on C can be represented as the difference of two convex
polynomials on C.
To see this, note that for sufficiently large λ>0 the polynomials p(x)+λ(x 2 1 +···+
x2 d ) and λ(x2 1 ···+x2 d ) both are convex on C by the convexity criterion of Brunn and
Hadamard. ⊓⊔
Proof (based on a theorem of Stone). The required result of Stone is as follows:
(1) Let S be a compact space and F a family of real continuous functions on S
having the following properties:
(i) F is closed under multiplication by real numbers, addition, and
multiplication.