Gruber P. Convex and Discrete Geometry

72 Convex Bodies
the existence of support hyperplanes can be extended to the case of unbounded proper
convex bodies such as the epigraph epi f of f .
The following simple result, essentially a version of Theorem 2.7, shows how
to move first-order differentiability results from convex functions to convex sets and
vice versa.
Lemma 5.1. Let D ⊆ E d be open and convex and f : D → R convex. Then, given
x ∈ D, the following are equivalent:
(i) f is differentiable at x.
(ii) � x, f (x) � is a smooth boundary point of epi f.
Proof. Left to the reader (use Theorem 2.7). ⊓⊔
Theorem 5.3. Let D ⊆ E d be open and convex and f : D → R convex. Then
f is differentiable at each point of D with a set of exceptions which is of σ -finite
(d − 1)-dimensional Hausdorff measure and meagre in D.
Proof. Using the same proofs, Theorems 5.1 and 5.2 easily extend to the graph of
f which is contained in the boundary of epi f in E d+1 . Now apply Lemma 5.1 and
note that subsets of bd epi f which are of σ -finite (d − 1)-dimensional Hausdorff
measure or are meagre project into such sets in D. ⊓⊔
This result is the analogue, for convex functions, of the Theorems 5.1 and 5.2
of Anderson and Klee and Mazur, respectively. For d = 1 the measure part says
that f has at most countably many points of non-differentiability, which was also
proved in Theorem 1.4. For general d, it is an essential refinement of Reidemeister’s
Theorem 2.6.
Second-Order Differentiability
When saying that the proper convex body C is twice or second-order differentiable at
a point x ∈ bd C, the following is meant: First, x is a smooth point of bd C. Next, let
H be the unique support hyperplane of C at x and u the interior unit normal vector of
C at x. Choose, in H, a Cartesian coordinate system with origin o at x. Together with
u it yields a Cartesian coordinate system in E d . In this coordinate system, represent
the lower side of bd C with respect to the last coordinate in the form
� y, g(y) � = y + g(y)u for y ∈ relint C ′ .
Here “ ′ ” is the orthogonal projection of E d onto H and g : relint C ′ → R a convex
function. We then require that, second, there is a positive semi-definite quadratic
form r on H such that
g(y) = r(y) + o(�y� 2 ) as y → o.
(Note that x is the origin.) Clearly, this definition may be extended to unbounded
proper convex bodies.
The following result makes it possible to transfer second-order differentiability
results from convex functions to convex bodies and vice versa.