Nonlinear Equations - UFRJ

[SEC. 6.1: LEGENDRE’S TRANSFORM 73
Lemma 6.2. A set U is convex if and only if U is an intersection
of closed half-spaces.
In order to prove this Lemma we need a classical fact about
Hilbert spaces:
Lemma 6.3. Let U be a convex subset in a Hilbert space, and let
p ∉ U. Then there is a hyperplane separating U and p, namely
where α ∈ E ∗ .
x ∈ U ⇒ α(x) < α(p)
This is a consequence of the Hahn-Banach theorem, see [23]
Lemma I.3 p.6.
Proof of Lemma 6.2. Assume that U is convex. Then, let S be the
collection of all half-spaces H α,α0 = {α(x)−α 0 ≥ 0}, α ∈ E ∗ , α 0 ∈ R,
such that U ⊆ H α,α0 .
Clearly
U ⊆
⋂
H α,α0 .
α,α 0∈S
Equality follows from Lemma 6.3.
The reciprocal is easy and left to the reader.
Definition 6.4. A function f : U ⊆ E → R is convex if and only if
its epigraph
Epi f = {(x, y) : f(x) ≤ y}
is convex.
Note that from this definition, the domain of a convex function
is always convex. In this book we shall convention that a convex
function has non-empty domain.
Definition 6.5. The Legendre-Fenchel transform of a function f :
U ⊆ E → R is the function f ∗ : U ∗ ⊆ E ∗ → R given by
f ∗ (α) = sup α(x) − f(x).
x∈U