Gruber P. Convex and Discrete Geometry

2 Convex Functions of Several Variables 37
(1) f � x, y(x) + s, y ′ (x) + t �
≥ f � x, y(x), y ′ (x) � � �
+ fy x, y(x), y ′ (x) s + fy ′
� �
x, y(x), y ′ (x) t
for (s, t) ∈ E2 .
Any function w :[a, b] →R of class C 1 with w(a) = α, w(b) = β can be represented
in the form w(x) = y(x) + s(x) for x ∈[a, b], where s :[a, b] →R is of
class C 1 and satisfies s(a) = s(b) = 0. Thus (1) and (ii) yield the following:
(2) I (w) = I (y + s) =
≥
�b
a
�
+
�b
f � x, y(x), y ′ (x) � dx
a
b
fy
�
= I (y) +
�
+
a
b
a
f � x, y(x) + s(x), y ′ (x) + s ′ (x) � dx
�
� � ′
x, y(x), y (x) s(x) dx +
a
b
�
d � � ′
fy ′ x, y(x), y (x)
dx �
s(x) dx
fy ′
� � ′ ′
x, y(x), y (x) s (x) dx
�b
d �
= I (y) + fy
dx
a
′
� � � ′
x, y(x), y (x) s(x) dx
= I (y) + fy ′
� � �
′ �
x, y(x), y (x) s(x) = I (y).
� b
a
a
b
fy ′
� � ′ ′
x, y(x), y (x) s (x) dx
Since w was arbitrary, this shows that y is a minimizer.
Assume, second, that f satisfies the condition of strict convexity. Then, in (1),
we have strict inequality unless (s, t) = (0, 0). (Otherwise the graph of the function
(s, t) → f � x, y(x)+s, y ′ (x)+t � and its affine support at (s, t) = (0, 0) have a linesegment
in common which contradicts the strict convexity.) Then strict inequality
holds in (2) unless s(x) = s ′ (x) = 0 for all x ∈[a, b], i.e. w = y. This shows that y
is the unique minimizer. ⊓⊔
The Background
In order to understand better what the result of Courant and Hilbert really means,
note the following: consider the (infinite dimensional) space of functions
F = � w :[a, b] →R, where w ∈ C 2 ,w(a) = α, w(b) = β � .