Gruber P. Convex and Discrete Geometry

2 Convex Functions of Several Variables 27
Theorem 2.8. Let C be open and f : C → R convex and differentiable on C. Then
all partial derivatives of f are continuous, i.e. f is of class C 1 .
Proof. It is sufficient to show the following:
(12) Let x, x1, x2, ···∈C be such that x1, x2, ···→ x. Then
un = grad f (xn) → grad f (x).
Theorem 2.7 implies that
(13) f (y) ≥ f (xn) + un · (y − xn) for y ∈ C and n = 1, 2,....
By Theorem 2.2, f is Lipschitz in a suitable neighborhood N ⊆ C of x with Lipschitz
constant L, say. By omitting finitely many xn and changing notation, if necessary,
we may assume that xn ∈ N for n = 1, 2,....For each n, choose yn ∈ N such
that yn − xn �= o and yn − xn has the same direction as un. These remarks, together
with (13), imply that
�un��yn − xn� =un · (yn − xn) ≤ f (yn) − f (xn) ≤ L�yn − xn�
and thus �un� ≤L, i.e. the sequence (un) is bounded. For the proof of (12) it is then
sufficient to show the following:
(14) Let (unk ) be a convergent subsequence of (un) with limit v, say. Then v =
grad f (x).
Since unk → v, xnk → x, and f is continuous, (13) implies that f (y) ≥ f (x) + v ·
(y − x) for each y ∈ C. Thus f is supported at x by the affine function a defined by
a(y) = f (x) + v · (y − x) for y ∈ Ed . Since f is differentiable at x, Theorem 2.7
implies that a is unique and v = grad f (x). ⊓⊔
Heuristic Observation
In Sect. 1.2 and in the present section we have encountered the phenomenon that a
convex function which has a particular property such as differentiability, has it in a
particularly pure form. This phenomenon occurs also in the context of the Venkov–
Alexandrov–McMullen theorem 32.2, 32.3 which shows that a convex polytope
which tiles by translation, is even a lattice tile. We think of these phenomena as
special cases of the following heuristic proposition.
Heuristic Principle. Consider a basic property which a convex function, a convex
body or a convex polytope can have. Then, in many cases, a convex function, body
or polytope which has this property, has an even stronger such property.
2.2 Alexandrov’s Theorem on Second-Order Differentiability
In the last section it was shown that a convex function is differentiable almost
everywhere, and we remarked that the same holds with respect to Baire category
and metric. A deep result of Alexandrov says that, in the sense of measure theory,