Gruber P. Convex and Discrete Geometry

8 The Brunn–Minkowski Inequality 149
inequality were given by Edler [285] (d = 2) and Schwarz [922] (d = 3). The latter
based his proof on the calculus of variations and used ideas of Weierstrass. For the
history of the classical isoperimetric problem, see Gericke [370] and Danilova [236].
As will be seen below, Minkowski surface area and the Brunn–Minkowski theorem
are ideal tools for an easy proof of the isoperimetric inequality for convex
bodies. The same proof holds also for much more general sets, but the isoperimetric
inequality thus obtained is not really of interest: Since the Minkowski surface area of
complicated sets in general is rather large, such sets readily satisfy the isoperimetric
inequality. For example, the Minkowski surface area of the set of rational points in
the unit cube is infinite, whereas for any reasonable notion of surface area, a countable
set should have area 0.
A good deal of the modern theory of geometric isoperimetric problems took place
outside convexity in areas where other notions of area measures were available, for
example, integral geometric concepts of area and the notions of perimeter and currents.
Instead of E d or S d−1 Riemannian and more general manifolds have been
considered.
Other developments related to the geometric isoperimetric inequality deal with
the so-called concentration of measure phenomenon in the context of the local theory
of normed spaces.
The following word of Poincaré applies well to the geometric isoperimetric
problem:
There are no solved problems, there are only more-or-less solved problems.
Below we reproduce Hurwitz’s [532] proof for d = 2 for Jordan domains,
by means of Fourier series and Minkowski’s [738] proof for general d for convex
bodies. Ingredients of Minkowski’s proof include Steiner’s formula for parallel bodies,
the Brunn–Minkowski inequality and Minkowski’s notion of surface area. For
Blaschke’s proof involving Steiner symmetrization, see Sect. 9.2.
Hurwitz’s Proof for Planar Jordan Curves by Means of Fourier Series
Hurwitz [532] derived the following version of the isoperimetric inequality. To simplify
the argument a little bit our assumptions are stronger than actually needed.
For more information on applications of Fourier series in convex geometry, see the
monographs of Groemer [405] and Koldobsky [606] and the proceedings on Fourier
analysis and convexity [343].
Theorem 8.6. Let K be a closed Jordan curve in E 2 of class C 2 with length L and
let A denote the area of the Jordan domain bounded by K . Then
L 2 ≥ 4π A,
where equality holds if and only if K is a circle.
Proof. We may assume that K is positively oriented, has length 2π, with parametrization
(x, y) :[0, 2π] →E 2 , where the parameter is the arc-length s. Since the
arc-length s is the parameter,