Gruber P. Convex and Discrete Geometry

6 Mixed Volumes and Quermassintegrals 91
The coefficient matrices of these systems of linear equations in a0n,...,adn, n =
1, 2,..., all are equal to a fixed Vandermonde matrix and thus non-singular. An
application of Cramer’s rule then shows that
Thus
a0n → a0,...,adn → ad, say.
�
a0 + a1λ +···+adλ
qn(λ) →
d
�
for each λ ≥ 0.
q(λ)
Hence q(λ) = a0 + a1λ +···+adλ d for λ ≥ 0, concluding the proof of (4).
We now prove the lemma by induction on m. In the following, all coefficients are
supposed to be symmetric in the indices. For m = 1letpn(λ) = anλ d . Then
an = pn(1) → p(1) = a, say,
by assumption. Thus
pn(λ) = anλ d �
a λd �
→ for each λ ≥ 0
p(λ)
by assumption. Hence p(λ) = aλ d for λ ≥ 0. (The case m = 1 is also a consequence
of Proposition (4).) Suppose now that m > 1 and that the lemma holds for
1, 2,...,m − 1. Then
pn(λ1,...,λm) = p0n(λ2,...,λm)+p1n(λ2,...,λm)λ1+···+pdn(λ2,...,λm)λ d 1 ,
where
pin(λ2,...,λm)
is a homogeneous polynomial of degree d − i in the variables λ2,...,λm. Given
λ2,...,λm ≥ 0, we have pn(λ1,λ2,...,λm) → p(λ1,λ2,...,λm) for any λ1 ≥ 0
by assumption. An application of (4) then shows that, for the coefficients pin, i =
0,...,d, (which are homogeneous polynomials in λ2,...,λm of degree d − i),
pin(λ2,...,λm) converges as n →∞for λ2,...,λm ≥ 0 and i = 0,...,d.
Denote the limit by qi(λ2,...,λm). The induction hypothesis, applied for i =
0,...,d, shows that
pin(λ2,...,λm) → qi(λ2,...,λm) for λ2,...,λm ≥ 0 and i = 0,...,d,
where qi(λ2,...,λm) is the restriction to λ2,...,λm ≥ 0 of a suitable homogeneous
polynomial in λ2,...,λm of degree d − i. Then, clearly,
pn(λ1,...,λd) =
d�
pin(λ2,...,λm)λ i 1 →
i=0
d�
i=0
qi(λ2,...,λm)λ i 1
= q(λ1,...,λm) for λ1 ≥ 0 and λ2,...,λm ≥ 0,
say, where q is a homogeneous polynomial of degree d in λ1,...,λm. Comparing
this with (3), we see that p(λ1,...,λm) = q(λ1,...,λm) for λ1,...,λm ≥ 0,
concluding the induction and thus the proof of the lemma. ⊓⊔