Gruber P. Convex and Discrete Geometry

90 Convex Bodies
If ei ∈ Pi is not extreme in Pi, then there is a line segment S ⊆ Pi such that ei is
a relative interior point of S. Then e is a relatively interior point of the line segment
λ1e1 +···+λi−1ei−1 + λi S + λi+1ei+1 +···+λmem ⊆ P and thus cannot be
extreme in P. This contradiction completes the proof of (1). Since by Theorem 5.5
each Pi has only finitely many extreme points, its vertices, (1) implies that P has only
finitely many extreme points. Since P is the convex hull of these by Theorem 5.5, it
is a polytope.
(ii) We show the following:
(2) Let Fi be a face of Pi for i = 1,...,m, and λ1,...,λm ≥ 0, such that
λ1F1 +···+λm Fm has dimension d − 1 (and is contained in a hyperplane
H). Then F1 +···+ Fm also has dimension d − 1 and is contained in a
hyperplane parallel to H.
It is sufficient to show this under the assumption that λi > 0 and o ∈ relint Fi for
each i. Then
H = lin(λ1F1 +···+λm Fm) = lin λ1F1 +···+lin λm Fm
= lin F1 +···+lin Fm = lin(F1 +···+ Fm),
concluding the proof of (2). Since each Pi has only finitely many faces, there is only
a finite set U of unit normal vectors of hyperplanes H as in (2). Since the hyperplanes
which contain facets of P = λ1 P1 +···+λm Pm are of this type by Lemma 6.1, the
proof of (ii) is complete. ⊓⊔
Lemma 6.3. Let pn, n = 1, 2,...,be a sequence of real homogeneous polynomials
in m variables of degree d and let p be a real function in m variables, defined for
λ1,...,λm ≥ 0. Assume that
(3) pn(λ1,...,λm) → p(λ1,...,λm) as n →∞
for each m-tuple λ1,...,λm ≥ 0.
Then p is a homogeneous polynomial in m variables of degree d, restricted to
λ1,...,λm ≥ 0.
Proof. First, the following will be shown:
(4) Let qn, n = 1, 2,...,be a sequence of real polynomials in one variable of
degree d and let q be a real function defined for λ ≥ 0. Assume that
qn(λ) → q(λ) for each λ ≥ 0.
Then q is a polynomial in one variable of degree d, restricted to λ ≥ 0.
Let qn(λ) = a0n + a1nλ +···+adnλ d for λ ∈ R. Then
a0n
= qn(0) → q(0),
a0n + a1n +···+adn = qn(1) → q(1),
a0n + a1n2 +···+adn2d = qn(2) → q(2),
..........................
a0n + a1nd +···+adndd = qn(d) → q(d).