Gruber P. Convex and Discrete Geometry

and thus
2 Convex Functions of Several Variables 25
f (y) ≥ f (x) + u · (y − x) + o(�y − x�) as y → x, y ∈ C.
Together with the inequality (7) this finally implies that
f (y) = f (x) + u · (y − x) + o(�y − x�) as y → x, y ∈ C. ⊓⊔
Simple examples show that in this result one may not replace convex by continuous.
The next result is due to Reidemeister [827].
Theorem 2.6. Let C be open and f : C → R convex. Then f is differentiable almost
everywhere on C.
Proof. We first show that
(10) The left-hand side and right-hand side partial derivatives f − xi , f + xi
1,...,d, exist on C and are measurable.
, i =
Let {b1,...,bd} be the standard basis of E d . For given i, consider the functions
gn, hn, n = 1, 2,...,which are defined by
By Theorem 1.4,
gn(x) = f (x − 1 n bi) − f (x)
− 1 n
hn(x) = f (x + 1 n bi) − f (x)
1
n
gn → f − xi , hn → f + xi
for x ∈ C such that x − 1
n bi ∈ C,
for x ∈ C such that x + 1
n bi ∈ C.
as n →∞on C.
As the pointwise limits of continuous functions, f − xi , f + xi
concluding the proof of (10).
Second,
(11) fxi , i = 1,...,d, exist almost everywhere on C.
For given i, theset � x ∈ C : f − xi (x) �= f + xi (x)�
are measurable on C,
is measurable on C by (10). Fubini’s theorem and Theorem 1.4 for convex functions
of one variable together imply that this set has measure 0, concluding the proof of
(11).
Having proved (11), Reidemeister’s theorem is an immediate consequence of
Theorem 2.5. ⊓⊔
A different proof of this result can be obtained from Theorems 2.2 and 2.5 and
Rademacher’s theorem on the differentiability almost everywhere of Lipschitz
continuous functions.