Gruber P. Convex and Discrete Geometry

204 Convex Bodies
For more information on John’s theorem and its aftermath, see the article of
Ball [53] and the surveys of Johnson and Lindenstrauss [552], Ball [54], and
Giannopoulos and Milman [374] in the Handbook of the Geometry of Banach
Spaces. For information on other John type and so-called minimum position results
see Gruber [445].
Uniqueness and John’s Characterization of Inscribed Ellipsoids of Maximum
Volume
Our aim here is to show the results of Löwner, Behrend [90], Danzer, Laugwitz and
Lenz [242], and of John [549], Pełczyński [788], and Ball [51], respectively, for
o-symmetric convex bodies.
Given d ×d matrices A = (aij), B = (bij), define their inner product by A· B =
� aijbij, where summation on i and j is from 1 to d. The dot · denotes also the
ordinary inner product in E d .Foru,v ∈ E d ,letu ⊗ v be the d ×d matrix u v T . Then
(u ⊗ v)x = (v · x) u for u,v,x ∈ E d . I is the d × d unit matrix.
Theorem 11.1. Let C ∈ Cp be symmetric in o. Then there is a unique ellipsoid of
maximum volume (necessarily symmetric in o) amongst all ellipsoids contained in C.
Theorem 11.2. Let C ∈ Cp be symmetric in o and B d ⊆ C. Then the following
statements are equivalent:
(i) Bd is the unique ellipsoid of maximum volume amongst all ellipsoids in C.
(ii) There are ui ∈ Bd ∩ bd C and λi > 0, i = 1,...,m, where d ≤ m ≤ 1 2d(d + 1)
such that
I = �
λi ui ⊗ ui, �
λk = d.
i
Here, and in the following, summation on i and k is from 1 to m. It is not difficult to
extend both theorems to convex bodies C which are not necessarily symmetric in o.
Without loss of generality, we assume, in the following, that all ellipsoids are
symmetric in o. Before beginning the proof, we state two tools:
(1) Each non-singular d ×d matrix M can be represented in the form M = AR,
where A is a symmetric, positive definite and R an orthogonal d ×d matrix.
(Take A = (MM T ) 1 2 and R = A −1 M, see [367], p.112.) Identify a symmetric d × d
matrix A = (aij) with the point (a11,...,a1d, a22,...,a2d,...,add) ∈ E 1 2 d(d+1) .
Then, the set of all symmetric positive definite d × d matrices is (represented by) an
open convex cone P in E 1 2 d(d+1) with apex at the origin. The set
(2) D ={A ∈ P : det A ≥ 1} is a closed, smooth and strictly convex set in P
with non-empty interior.
This follows from the implicit function theorem and Minkowski’s determinant
inequality,
k