Gruber P. Convex and Discrete Geometry

36 Convex Functions
x x
Fig. 2.1. Stationary, extreme and unique extreme points
local minimum at a point x ∈ int I, necessarily f ′ (x) = 0. Conversely, the condition
that f ′ (x) = 0 does not guarantee that f attains a local minimum at x, but together
with the condition that f is convex, it does. If, in addition, we know that f is strictly
convex, this local minimum is even the unique global minimum of f (Fig. 2.1).
A similar situation arises in the calculus of variations: Consider a variational
problem and the corresponding Euler–Lagrange equation(s). A smooth solution of
the variational problem necessarily satisfies the Euler–Lagrange equation(s). Conversely,
if a function satisfies the Euler–Lagrange equation(s), this may not be sufficient
for the function to be a solution of the variational problem. But it is sufficient
if, in addition, certain convexity conditions are satisfied. The first result of this type
seems to be due to Courant and Hilbert [227], p.186, which, in essence, is reproduced
below.
For a wealth of more recent pertinent results and references, see the survey of
Brechtken–Manderscheid and Heil [165].
A Sufficient Condition
A result of Courant and Hilbert is as follows, where [a, b] is an interval in R.
Theorem 2.12. Let f :[a, b]×E 2 → R be of class C 2 and assume that for each fixed
x ∈[a, b] the function (y, z) → f (x, y, z) for (y, z) ∈ E 2 is convex, respectively,
strictly convex. Let α, β ∈ R and assume that y :[a, b] →R is a function of class
C 1 such that y(a) = α, y(b) = β. Then the following statements are equivalent:
(i) y is a minimizer, respectively, unique minimizer of the integral
I (w) =
�b
a
f � x,w(x), w ′ (x) � dx
among all functions w :[a, b] →R of class C 1 with w(a) = α, w(b) = β.
(ii) y satisfies the Euler–Lagrange equation
fy(x, y, y ′ ) − d
dx fy ′(x, y, y′ ) = 0.
Proof. (i)⇒(ii) This is a standard result in the calculus of variations.
(ii)⇒(i) Assume first, that f satisfies the convexity condition. Then, for each
x ∈[a, b], the expression f � x, y(x) + s, y ′ (x) + t � is convex in (s, t) on E 2 and
Theorem 2.7 implies that, in particular,
x