Gruber P. Convex and Discrete Geometry

18 Theorems of Alexandrov, Minkowski and Lindelöf 309
(1) Let P, Q ∈ Pp have the same set of exterior normal vectors of their facets
and such that Q is circumscribed to B d . Then
S(P) d S(Q)d
≥
V (P) d−1 V (Q)
d−1 ,
where equality holds if and only if P is homothetic to Q (and thus also
circumscribed to a ball).
Since Q is circumscribed to the unit ball B d , dissecting Q into pyramids, all with
apex o, the formula for the volume of a pyramid then shows that
(2) S(Q) = dV(Q).
Minkowski’s theorem on mixed volumes 6.5 yields
(3) V � (1 − λ)P + λQ � is a polynomial in λ for 0 ≤ λ ≤ 1.
Q is circumscribed to B d . Choose ϱ>0 such that Q ⊆ ϱB d .For0≤ λ ≤ 1the
polytope (1−λ)P +λQ can thus be dissected into (1−λ)P, right prisms of height λ
with the facets of (1 − λ)P as bases, and a set in the λϱ-neighbourhood of the union
of the (d − 2)-faces of (1 − λ)P. Thus
(4) V � (1 − λ)P + λQ � = (1 − λ) d V (P) + (1 − λ) d−1 λS(P) + O(λ 2 )
as λ →+0.
The Brunn–Minkowski theorem 8.3 shows that
(5) the function f (λ) = V � (1 − λ)P + λQ � 1 d − (1 − λ)V (P) 1 d − λV (Q) 1 d
for 0 ≤ λ ≤ 1 with f (0) = f (1) = 0 is strictly concave, unless P is
homothetic to Q, in which case it is identically 0.
By (3) f is differentiable. Thus (5) shows that
(6) f ′ (0) ≥ 0 where equality holds if and only if P is homothetic to Q.
Using the definition of f in (5), (4), and (2), a calculation which is almost identical
to that in the proof of the isoperimetric theorem 8.7, yields (1). ⊓⊔
Corollary 18.2. Among all proper convex polytopes in E d with a given number of
facets, there are polytopes with minimum isoperimetric quotient and these polytopes
are circumscribed to a ball.
In the proof, the first step is to show that there is a polytope with minimum isoperimetric
quotient. By Lindelöf’s theorem this polytope then is circumscribed to a ball.
Remark. Diskant [274] extended Lindelöf’s theorem to the case where ordinary surface
area is replaced by generalized surface area, see Sect. 8.3. The above corollary
and its generalization by Diskant are used by Gruber [439,443] to obtain information
about the geometric form of convex polytopes with minimum isoperimetric quotient.