Gruber P. Convex and Discrete Geometry

11 Approximation of Convex Bodies and Its Applications 203
Hausdorff metric δ H and then by Gruber [427] for the symmetric difference metric
δ V . These results gave rise to a long series of further investigations.
Early contributions to random approximation are due to Crofton, Czuber and
Blaschke, the first asymptotic results were proved by Rényi and Sulanke [831] in
case d = 2. Modern results for general d are due, amongst others, to Bárány and
Buchta [69] and Reitzner [829].
In this section we first consider John’s characterization of the ellipsoid of maximum
volume contained in a convex body and use it to prove Ball’s reverse
isoperimetric inequality. Then the author’s asymptotic formula for the best volume
approximation of a convex body by circumscribed convex polytopes is given
as the number of facets tends to infinity. This is then applied to the isoperimetric
problem for convex polytopes.
For references concerning John’s theorem and asymptotic best approximation,
see the following sections. Recent surveys on random approximation are due to the
author [435] and Schneider [909].
11.1 John’s Ellipsoid Theorem and Ball’s Reverse Isoperimetric Inequality
To a given convex body one may assign several ellipsoids in a canonical way.
One example is the ellipsoid of inertia, for other examples, see the articles of
Milman [725] and Lutwak, Yang and Zhang [670] and the book of Pisier [802].
Among the ellipsoids which are inscribed, respectively, circumscribed to a proper
convex body in E d , there is precisely one of maximum, respectively, minimum volume.
Simple proofs for this result are due to Löwner (unpublished), Behrend [90]
(d = 2) and Danzer, Laugwitz and Lenz [242] (general d). John [549] characterized
inscribed ellipsoids of maximum volume, a complement being due to Ball [51]. Both
results are of interest in convex geometry and in the geometry of normed spaces.
John’s theorem implies, in particular, that for any origin symmetric convex body
there is an ellipsoid which approximates it up to a factor √ d.
Below these results are proved for convex bodies which are symmetric in o. The
proof of the characterization result is taken from Gruber and Schuster [452]. It is
based on the idea of Voronoĭ to identify symmetric, positive definite d × d matrices,
respectively, positive definite quadratic forms on E d with points in E 1 2 d(d+1) ,
compare Sect. 29.4. This idea was applied earlier in the same context by the author
[421]. We give two classical applications, one to the group of affinities which map
a convex body onto itself and one to the Banach–Mazur distance between norms on
E d . A third application is the reverse isoperimetric inequality. For convex bodies C,
the isoperimetric quotient
S(C) d
V (C) d−1
is bounded below by the isoperimetric quotient of the unit ball B d , but it is clearly not
bounded above. Behrend [89], for d = 2, and Ball [50], for general d, asked whether
for any given convex body C there is an affine image, the isoperimetric quotient of
which is bounded above in terms of d, and what is the worst case.