Gruber P. Convex and Discrete Geometry

308 Convex Polytopes
18.3 The Isoperimetric Problem for Convex Polytopes and Lindelöf’s Theorem
Considering convex polytopes, the following natural isoperimetric problem arises.
Determine, among all proper convex polytopes in E d with n facets, those with
minimum isoperimetric quotient and specify the value of the latter.
This section contains Lindelöf’s necessary condition for polytopes with minimum
isoperimetric quotient amongst all convex polytopes in E d with n facets.
Pertinent results in E 3 , for small values of n, are reviewed by Fejes Tóth [330]
and Florian [337]. This includes characterizations of regular polytopes. For asymptotic
results about the minimum isoperimetric quotient and the form of the minimizing
polytopes as n →∞, see Gruber [439–441, 443]. Compare also Sects. 8.3, 8.6
and 9.2.
Lindelöf’s Necessary Condition for Minimum Isoperimetric Quotient
The following result of Lindelöf [658] was vaguely anticipated by Steiner [960]. The
proof given below is modeled along the lines of the proof of the classical isoperimetric
theorem 8.7.
Theorem 18.4. Among all proper convex polytopes in E d with given exterior normals
of the facets, it is precisely the polytopes circumscribed to a ball that have
minimum isoperimetric quotient (Fig. 18.2).
The proof which we present in the following makes use of Minkowski’s theorem
on mixed volumes and the Brunn–Minkowski theorem. A different proof, which is
left to the reader, is based on Minkowski’s first inequality for mixed volumes. These
proofs are very similar to the second and third proof of the isoperimetric inequality,
see Theorem 8.7.
Proof. Since homotheties do not change the isoperimetric quotient of a convex body,
it is sufficient to show the following:
P
S(P) d
V (P) d−1 ≥ S(Q)d
V (P) d−1
Fig. 18.2. Lindelöf’s isoperimetric theorem for polytopes
o
Q
B 2