Gruber P. Convex and Discrete Geometry

5 The Boundary of a Convex Body 73
Lemma 5.2. Let D ⊆ E d be open and convex and f : D → R convex. Then, given
x ∈ D, the following statements are equivalent:
(i) f is twice differentiable at x in the sense of Theorem 2.9,
(ii) epi f is twice differentiable at � x, f (x) � .
Proof. (i)⇒(ii) We may suppose that x = o, f (o) = 0. By assumption, there are a
vector a ∈ E d and a positive semi-definite quadratic form q(z) = z T Az for z ∈ E d ,
such that
(7) f (z) = a · z + q(z) + o(�z� 2 ) as z → o (= x), z ∈ E d .
Let v be the unit normal vector of E d pointing into int epi f .Let“ ′ ” denote the
orthogonal projection of E d+1 = E d × R onto E d . Since, in particular, f is differentiable
at o, Lemma 5.1 shows that o is a smooth point of bd epi f .LetH be the
support hyperplane of epi f at o. Denote by u its unit normal vector pointing into
int epi f .Let“ ′′ ” denote the orthogonal projection of E d+1 onto H. Represent the
lower side of bd epi f , with respect to the last coordinate of the Cartesian coordinate
system determined by H, and u in the form
� y, g(y) � = y + g(y)u for y ∈ (relint epi f ) ′′ ⊆ H.
Since o is a smooth point of bd epi f , Lemma 5.1 shows that the convex function g
is differentiable at o and thus (Fig. 5.2)
(8) g(y) = o(�y�) as y → o, y ∈ H.
Combining (7), z = (y + g(y)u) ′ = y ′ + g(y)u ′ and (8), we see that
g(y) = � f (z) − a · z � (u · v) = � q(z) + o(�z� 2 ) � (u · v)
= � q(y ′ ) + 2y ′T Au ′ g(y) + q(u ′ )g(y) 2 + o(�y� 2 ) � (u · v)
= r(y) + o(�y� 2 ) as y → o,
u
v
R
y
o y ′ z
g(y)
H
f (z)
Fig. 5.2. Twice differentiability of functions and epigraphs
E d