Gruber P. Convex and Discrete Geometry

18 Theorems of Alexandrov, Minkowski and Lindelöf 305
(2) V (s) = V � P(s) � is differentiable for s1,...,sn > 0, and
∂V (s)
= Ai(s) for i = 1,...,n.
∂si
The proof of the formula for the partial derivatives of V (s), in the case where
Ai(s) = 0, is left to the reader. If Ai(s) >0, then, for sufficiently small |h|, the
polytopes P(s1,...,si + h,...,sn) and P(s1,...,sn) differ by the convex hull of
Fi(s1,...,si + h,...,sn) and Fi(s1,...,sn). Since Fi(s1,...,si + h,...,sn) tends
to F(s1,...,sn) as h → 0 (with respect to the Hausdorff metric), the volume of the
convex hull is Ai(s1,...,sn)|h|+o(|h|). Hence
V (s1,...,si + h,...,sn) − V (s1,...,sn) = A(s1,...,sn)h + o(h) as h → 0.
Now divide by h and let h tend to 0 to get the formula for the partial derivatives of
V (s) in (2). From the formula and the continuity of Ai, the differentiability of V
follows.
The simplex
�
n�
�
(3) S = s : si ≥ 0, αisi = 1 ⊆ E n
i=1
is compact. Since V is continuous on S, it attains its maximum on S at a point of
S, saya. Ifa �∈ relint S, at least one ai is 0 and thus o ∈ bd P(a) by (1). Choose a
vector v �= o such that o ∈ int � P(a) + v � . Then
P(a) + v = � x + v : ui · x ≤ ai, i = 1,...,n �
= � y : ui · y ≤ ai + ui · v, i = 1,...,n �
= P(b), where b = a + (u1 · v,...,un · v).
Since o ∈ int � P(a) + v � = int P(b), we thus see that b1,...,bn > 0. Since a ∈ S,
the equality in statement (ii) implies that
n�
i=1
αibi =
n�
i=1
αiai +
Hence b ∈ relint S. We thus have shown that
n�
αi ui · v = 1.
i=1
V attains its maximum on S at the point b ∈ relint S.
Propositions (2), (3) and the Lagrange multiplier theorem then yield
�
n�
� ����s=b
∂
V (s) − λ αisi = Ai(b) − λαi = 0fori = 1,...,n,
∂si
i=1