Gruber P. Convex and Discrete Geometry

148 Convex Bodies
In the following we present the classical geometric isoperimetric and the isodiametric
inequality. Then we consider an additional norm on E d . In the normed space
thus obtained, we define the notion of generalized surface area and state the corresponding
isoperimetric inequality.
For information on notions of surface area in geometric measure theory and the
corresponding isoperimetric problems, including isoperimetric problems on manifolds,
see Burago and Zalgaller [178], Talenti [986], Morgan [756], Chavel [201]
and Ritoré and Ros [839]. These notions and problems are not touched here.
The basic reference for generalized surface area in finite-dimensional normed
spaces and related matters is Thompson’s monograph [994]. For pertinent material
in the context of the local theory of normed spaces, see Sect. 8.6 and the references
cited there, in particular the book of Ledoux [634]. Closer to convex geometry is
Ball’s [50] reverse isoperimetric inequality, see Theorem 11.3.
The Classical Isoperimetric Inequality
A common version of the isoperimetric inequality says that, amongst all convex bodies
in E d of given volume, it is precisely the Euclidean balls that have minimum
surface area.
The planar case of this problem is mentioned in the Aeneid of Vergil [1009],
book 1, verses 367, 368: Dido fled with a group of followers from Tyrus to escape
the fate of her husband Sychaeus, who was killed by her brother Pygmalion. Close
to where now is Tunis
they bought as much land – and called it Byrsa – as could be encircled with a bull’s
hide.
Geometric results in antiquity concerning the isoperimetric problem are due to
Archimedes (lost, but referred to by other authors) and Zenodorus. The contributions
of Galilei in the renaissance are in the spirit of Zenodorus, while, in the eighteenth
century, analytic attempts to prove the isoperimetric inequality drew upon tools from
the calculus of variations. More geometric were the four hinge method and Steiner
symmetrization, introduced to the isoperimetric problem in the late eighteenth and
the first half of the nineteenth century. The arguments of the analysts and the geometric
arguments of Steiner in the nineteenth century to prove the isoperimetric inequality
all made use of the implicit unproven assumption that there was a solution
and thus did not lead to a complete solution, as noted by Dirichlet.
Steiner, in particular, proved that a planar convex body which is not a circle is
not a solution of the isoperimetric problem. He thought that this implies that the
circular discs were the solutions. To show his error more clearly, we present Pólya’s
[810] pseudo-argument for the assertion that 1 is the largest positive integer: For any
positive integer n > 1 there are integers larger than n, for example n 2 . Thus n cannot
be the largest positive integer. The only possible candidate thus is 1. Therefore 1 is
the largest positive integer.
The importance of Steiner, in the context of the isoperimetric problem, lies in
the methodological ideas. The first complete proofs of the geometric isoperimetric