Gruber P. Convex and Discrete Geometry

28 Convex Functions
even more is true: a convex function is twice differentiable, except on a set of
measure zero. In contrast, results of Zamfirescu [1037, 1038] imply that, from the
Baire category viewpoint, a typical convex function is twice differentiable only on a
small set.
This section contains a proof of the theorem of Alexandrov [14] on second order
differentiability of convex functions. It generalizes, in a non-trivial way, a result of
Busemann and Feller [183] for d = 2 and, of course, Theorem 1.7 for d = 1.
Besides the original proof of Alexandrov, we mention a proof of Zajíček [1036]
and one in the book of Evans and Gariepy [314].
For more information on differentiability properties of convex functions, respectively,
smooth boundary points of convex bodies, see Schneider [904, 907, 908] and
Gruber [431].
Second-Order Differentiability
A function f : C → R is twice (or second-order) differentiable at a point x ∈ int C,
if there are a vector u ∈ E d ,thegradient of f at x, and a real d × d matrix H, the
Hessian matrix of f at x, such that
f (y) = f (x) + u · (y − x) + 1
2 (y − x)T H(y − x) + o(�y − x� 2 ) as y → x.
Alexandrov’s Theorem
Following Zajíček [1036], we prove Alexandrov’s differentiability theorem [14]:
Theorem 2.9. Let C be open and f : C → R convex. Then f is twice differentiable
almost everywhere on C.
Because the required tools are cited explicitly and the necessary definitions and
explanations are incorporated into the proof, the latter looks longer than it actually
is. A source for the tools is Mattila [696].
Proof. First, several tools are collected. The extension theorem of McShane shows
that
(1) A Lipschitz mapping which maps a set D ⊆ E d into E d can be extended to
a Lipschitz mapping of E d into E d with the same Lipschitz constant.
By Rademacher’s differentiability theorem,
(2) A Lipschitz mapping which maps C into E d is almost everywhere differentiable
on C.
Call a mapping K : C → E d differentiable (in the sense of Stolz or Fréchet) at a
point x ∈ C if there is a real d × d matrix A, thederivative of K at x, such that
K (y) = K (x) + A(y − x) + o(�y − x�) as y → x, y ∈ C.
The next result is a version of Sard’s theorem.