Gruber P. Convex and Discrete Geometry

32 Convex Functions
by (15), uniformly for all h. Hence
(16) f (y) = f (x) + u · (y − x) + 1
2 (y − x)T H(y − x) + o(�y − x� 2 ) as
y → x, y ∈ C,
uniformly for y of the form x + th and thus for almost all y ∈ C. Since f (y) and
f (x) + u · (y − x) + 1 2 (y − x)T H(y − x) depend continuously on y, (16) holds for
all y ∈ C. This concludes the proof of (14) and thus of Alexandrov’s theorem. ⊓⊔
In contrast to the fact established in Theorem 2.8 that the existence of all first partial
derivatives of a convex function implies that the function is of class C 1 , the existence
of all second partial derivatives or the second order differentiability of a convex function
does not guarantee that it is of class C 2 .
2.3 A Convexity Criterion
As for functions of one variable, it is of interest to ascertain when a function f :
C → R of several variables is convex.
Below we give a simple, yet useful convexity criterion due to Brunn [174] and
Hadamard [460].
Hessians
Let f : C → R have partial derivatives of second order, fxi ,xk (x), i, k = 1,...,d,
at x ∈ C. As in Section 2.2, the d × d matrix
⎛
⎞
H = H(x) =
fx1,x1
(x) ... fx1,xd (x)
⎜ fx2,x1 ⎜ (x) ... fx2,xd (x)
⎝ ....................
fxd ,x1 (x) ... fxd ,xd (x)
⎟
⎠
is called the Hessian matrix of f at x. Corresponding to it is the Hessian (quadratic)
form of f at x, defined by
y → 1
2 yT Hy = 1
2
d�
i,k=1
The Convexity Criterion of Brunn and Hadamard
fxi ,xk (x) yi yk for y ∈ E d .
The following result for convex functions of d variables will be proved by retracing
it back to the one-dimensional case and using the fact that a convex function in one
variable of class C 2 has non-negative second derivative.
Theorem 2.10. Let C be open and f : C → R of class C 2 . Then the following
statements are equivalent:
(i) f is convex.
(ii) For any x ∈ C the Hessian form H(x) of f at x is positive semi-definite.