Gruber P. Convex and Discrete Geometry

74 Convex Bodies
where r(·) is a suitable positive semi-definite quadratic form on H.
(ii)⇒(i) Let x = o, f (o) = 0, u,v,H, r, ′′ , g be as before. By assumption,
(9) g(y) = r(y) + o(�y� 2 ) as y → o, y ∈ H,
where r(y) = y T By,fory ∈ H, is a positive semi-definite quadratic form on H.
Lemma 5.1 shows that f is differentiable at o, that is
(10) f (z) = a · z + o(�z�) as z → o, z ∈ D,
where a is a suitable vector in E d . Note that
(11) y = � z + f (z)v � ′′ = z ′′ + f (z)v ′′ .
Propositions (9)–(11) yield the following:
and then
f (z) = � y + g(y)u � · v = y · v + g(y) u · v
= z ′′ · v + f (z)v ′′ · v + g � z ′′ + f (z)v ′′� u · v,
f (z)(1 − v ′′ · v) = z ′′ · v + r � z ′′ + f (z)v ′′� u · v + o � �z ′′ + f (z)v ′′ � 2� u · v
= z ′′ · v + � z ′′T Bz ′′ + z ′′T Bv ′′ 2 f (z) + v ′′T Bv ′′ f (z) 2� u · v + o(�z� 2 )
= z ′′ · v + � z ′′T Bz ′′ + z ′′T Bv ′′ 2 a · z + v ′′T Bv ′′ (a · z) 2� u · v + o(�z� 2 )
= b · z + q(z) + o(�z� 2 ) as z → o, z ∈ D.
Here b and q are a suitable vector in E d and a quadratic form on E d . Since f is
convex, this can hold only if q is positive semi-definite. ⊓⊔
Using Lemma 5.2 and Alexandrov’s Theorem 2.9 for convex functions, we obtain
Alexandrov’s theorem on second-order differentiability of convex bodies:
Theorem 5.4. Let C ∈ Cp. Then bd C is almost everywhere twice differentiable.
Proof. bd C can be covered by finitely many relative interiors of its lower sides with
respect to suitable Cartesian coordinate systems. To the corresponding convex functions,
apply Alexandrov’s theorem 2.9 and Lemma 5.2, noting that a set on the relative
interior of a lower side of bd C has (d − 1)-dimensional Hausdorff measure
0 if its orthogonal projection into a hyperplane has (d − 1)-dimensional Hausdorff
measure 0. (A convex function on an open set is locally Lipschitz.) ⊓⊔
5.2 Extreme Points
Extreme points of convex bodies play an important role in convex analysis, convex
geometry and functional analysis, for example in the context of the Krein–Milman
theorem and Choquet theory. A refinement of the notion of extreme point is that of
exposed point. Using this notion, Straszewicz [973] proved a result of Krein–Milman
type.