Page 55 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

42 Preliminaries

We now gather some properties of the Legendre transform (for a proof see
the exercises).

n
n
Theorem 1.54 Let f : R → R (or f : R → R ∪ {+∞}).
∗
(i) The function f is convex (even if f is not).
(ii) The function f ∗∗ is convex and f ∗∗ ≤ f. If, furthermore, f is convex
and finite then f ∗∗ = f.More generally, if f is finite but not necessarily convex,
then f ∗∗ is its convex envelope (which means that it is the largest convex function
that is smaller than f).

(iii) The following identity always holds: f ∗∗∗ = f .
∗
1
n
(iv) If f ∈ C (R ),convexand finite, then
n
∗
f (x)+ f (∇f (x)) = h∇f (x); xi , ∀x ∈ R .
n
(v) If f : R → R is strictly convex and if
f (x)
lim =+∞
|x|→∞ |x|
n
1
n
1
then f ∈ C (R ).Moreover if f ∈ C (R ) and
∗
∗ ∗ ∗
f (x)+ f (x )= hx ; xi
then
∗
∗
x = ∇f (x) and x = ∇f (x ) .
∗
We finally conclude with a theorem that allows to compute the convex en-
velope without using duality (see Theorem 2.2.9 in [31] or Corollary 17.1.5 in
Rockafellar [87]).

n
Theorem 1.55 (Carathéodory theorem). Let f : R → R then
( )
n+1 n+1 n+1
X X X
f ∗∗ (x)= inf λ i f (x i ): x = λ i x i ,λ i ≥ 0 and λ i =1 .
i=1 i=1 i=1