Gruber P. Convex and Discrete Geometry

11 Approximation of Convex Bodies and Its Applications 207
The number of common points of bd C and the inscribed ellipsoid of maximum
volume is precisely 1 2d(d + 1) for most convex bodies C which are symmetric in o.
See Gruber [421] and, for a different proof, Rudelson [860].
The Group of Affinities of a Convex Body
Danzer, Laugwitz and Lenz [242], proved the following result.
Corollary 11.1. Let C ∈ Cp be symmetric in o. Then the group L of all linear
transformations which map C onto itself is a subgroup of the orthogonal group with
respect to a suitable inner product.
Proof. By replacing the given inner product on E d by a suitable new inner product, if
necessary, we may assume that the unique ellipsoid of maximum volume in C is the
unit ball with respect to the new inner product. Denote the latter by B d .LetT ∈ L.
Since TC = C, the transformation T is volume-preserving. Thus TB d is also an
ellipsoid of maximum volume in TC = C. The uniqueness part of Theorem 11.1
then shows that TB d = B d , i.e. T is an orthogonal transformation with respect to
the inner product corresponding to B d . ⊓⊔
Remark. A similar result holds for general convex bodies and affinities.
The Banach–Mazur Compactum
On the space N = N (E d ) of all norms on E d define the Banach–Mazur distance
δ BM as follows, where for a norm |·|the corresponding unit ball is denoted by B|·|.
δ BM (|·|, ⌊⌉·⌊⌉) = inf � λ>1 :∃T : E d → E d �
linear, B|·| ⊆ TB⌊⌉·⌊⌉ ⊆ λB|·|
for |·|, ⌊⌉·⌊⌉∈N .
δ BM does not distinguish between isometric norms, is symmetric and log δ BM satisfies
the triangle inequality. There are other ways to define δ BM . N , endowed with
the distance δ BM , is a compact space, called the Banach–Mazur compactum. It has
attracted a lot of interest in the local theory of normed spaces, see the reports of
Gluskin [382] and Szarek [981] and the book of Tomczak-Jaegermann [1001]. One
of the difficult open questions in this area is to determine the diameter of N . Here,
the following simple estimate due to John [549] is given, where �·�denotes the
Euclidean norm on E d .
Corollary 11.2. δ BM (�·�, |·|) ≤ √ d for any |·|∈N (E d ).
Proof. Assume that T is chosen such that Bd ⊆ TB|·| is the ellipsoid of maximum
volume in C = TB|·|. Choose ui,λi according to John’s ellipsoid theorem. Then
(6) I = �
λi ui ⊗ ui, �
λi = d.
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