Gruber P. Convex and Discrete Geometry

Proof. (i)⇒(ii) It is sufficient to show the following:
2 Convex Functions of Several Variables 33
(1) Let x, y ∈ C, x �= y. Then (y − x) T H(x)(y − x) ≥ 0.
Let g be the convex function defined by g(t) = f � (1 − t)x + ty � for t ∈ R such that
(1 − t)x + ty = x + t(y − x) ∈ C. Since f and thus g is of class C 2 , the chain rule
for functions of d variables yields the following:
g ′ (t) exists and equals
g ′′ (t) exists and equals
d�
i=1
d�
i,k=1
fxi
� x + t(y − x) � (yi − xi),
fxi ,xk
� x + t(y − x) � (yi − xi)(yk − xk).
Since g is convex and of class C 2 , Corollary 1.1 implies that, in particular, g ′′ (0) ≥ 0
and thus
1
2 (y − x)T H(x)(y − x) = 1
2
d�
i,k=1
fxi ,xk (x)(yi − xi)(yk − xk) = 1
2 g′′ (0) ≥ 0,
concluding the proof of (1).
(ii)⇒(i) By Theorem 2.4, it is sufficient to show the following:
(2) Let x ∈ C. Then f has an affine support at x.
Since f is of class C 2 , Taylor’s theorem, for functions of d variables, implies that
f (y) = f (x) + u · (y − x) + 1
2
d�
i,k=1
fxi ,xk
� x + ϑ(y − x) � (yi − xi)(yk − xk)
= f (x) + u · (y − x) + 1
2 (y − x)T H � x + ϑ(y − x) � (y − x)
≥ f (x) + u · (y − x) for y ∈ C,
where u = grad f (x), 0