Gruber P. Convex and Discrete Geometry

30 Convex Functions
G(x). The proof of this statement is verbatim the same as for the corresponding
statement (12) in the proof of Theorem 2.8.
Fourth, call G differentiable at x ∈ C if G(x) is single-valued at x,sayG(x) = u,
and if there is a real d × d matrix H, thederivative of G at x, such that the set
G(y) − u − H(y − x) is contained in a neighborhood of o of radius o(�y − x�)
as y → x, y ∈ C. The latter property will also be expressed in the form v =
u + H(y − x) + o(�y − x�) as y → x, y ∈ C uniformly for v ∈ G(y),orintheform
G(y) = u + H(y − x) + o(�y − x�) as y → x, y ∈ C.Clearly,ifG is differentiable
at x, then it is continuous at x.
(9) Let G be continuous at x ∈ C and let the single-valued mapping K =
(I + G) −1 : D → E d be differentiable at w = x + u where u = G(x), and
such that its derivative A is non-singular. Then G is differentiable at x with
derivative H = A −1 − I .
The differentiability of K at w implies that
(10) K (z) − K (w) = A(z − w) + r for z ∈ D, where �r�/�z − w� is arbitrarily
small if �z − w� is sufficiently small.
For a real d × d matrix B = (bik) define �B� =( � b 2 ik ) 1 2 . A result from linear
algebra based on the Cauchy–Schwarz inequality then shows that �Bp�≤�B��p�
for p ∈ E d . Thus �p� =�B −1 Bp�≤�B −1 ��Bp�, or
(11) �Bp�≥ �p�
�B −1 � for p ∈ Ed and each non-singular d × d matrix B.
The following statement is a consequence of the continuity of G at x.
(12) Let w = x +u ∈ D, u = G(x), z = y +v ∈ D, y ∈ C,v ∈ G(y) and thus
x = K (w), y = K (z). Then �z − w� (≤ �v − u�+�y − x�) is arbitrarily
small, uniformly for v ∈ G(y), if�y − x� is sufficiently small.
For x, y,w,z as in (12), Propositions (10)–(12) show that
�y − x� =�K (z) − K (w)� ≥�A(z − w)�−�r� ≥
= �z − w�
2�A −1 �
if �y − x� is sufficiently small.
�z − w�
�A −1 �
− �z − w�
2�A −1 �
Combining this with (10) yields the following proposition and thus completes the
proof of statement (9):
Let w = x + u ∈ D, u = G(x), z = y + v ∈ D, y ∈ C,v ∈ G(y) and
thus x = K (w), y = K (z). Then z − w = A −1 (y − x) + A −1 r,or
v = u − (y − x) + A −1 (y − x) + A −1 r
= u + (A −1 − I )(y − x) + A −1 r,
where �A −1 r�/�y − x� is arbitrarily small, uniformly for v ∈ G(y), if
�y − x� is sufficiently small.