Gruber P. Convex and Discrete Geometry

26 Convex Functions
Remark. More precise measure-theoretic information on the size of the set of points
at which a convex function is differentiable was given by Anderson and Klee [28]
(in the guise of a result on convex bodies). Mazur [701] and Preiss and Zajíček
[815, 816] employ the topological tool of Baire categories, respectively, the metric
tool of porous sets to estimate the size of this differentiability set. From all three
points of view, it is a large set. Compare Theorems 5.1 and 5.2 dealing with convex
bodies and the discussion on first-order differentiability after these results. For a
discussion of differentiability properties of convex functions on infinite dimensional
spaces, see the book of Benyamini and Lindenstrauss [97].
First-Order Differentiability and Affine Support
Our first result is as follows:
Theorem 2.7. Let f : C → R be convex and x ∈ int C. Then the following are
equivalent:
(i) f is differentiable at x.
(ii) f has unique affine support at x, say a : E d → R, where a(y) = f (x)+u·(y−x)
for y ∈ E d and u = grad f (x) = � fx1 (x), . . . , fxd (x)� .
Proof. By Theorem 1.4,
f − xi (x) and f + xi
(x), i = 1,...,d, exist.
(i)⇒(ii) By Theorem 2.4 f has an affine support a : E d → R at x, where
a(y) = f (x) + u · (y − x) for y ∈ E d .Leti = 1,...,d. The restriction of f to
the intersection of C with the line through x parallel to the ith coordinate axis is a
convex function of one variable. This function has derivative
fxi (x) = f − xi (x) = f + xi (x)
at x by Theorem 2.5 and f (x)+ui (yi − xi) for yi ∈ R is an affine support at x. Thus
ui = fxi (x) by Proposition 1.2. Since this holds for i = 1,...,d, the affine support
a is unique and has the desired form.
(ii)⇒(i) If f is not differentiable at x, then there is an index i such that fxi (x)
does not exist, see Theorem 2.5. Then
f − xi (x) < f + xi (x)
by Theorem 1.4. By Proposition 1.2, each affine function of the form f (x) + ui(yi −
xi) for yi ∈ R where
f − xi (x) ≤ ui ≤ f + xi (x)
is an affine support of the restriction of f to the intersection of C and the line through
x parallel to the ith coordinate axis. Each of these affine supports can be extended to
an affine support of f at x by the Hahn–Banach type theorem 2.3. Hence f does not
have a unique affine support at x. ⊓⊔