Gruber P. Convex and Discrete Geometry

11 Approximation of Convex Bodies and Its Applications 205
� det(A+ B) � 1 d ≥ � det A � 1 d + � det B � 1 d for A, B ∈ P, where equality holds
precisely in case where A, B are proportional.
A proof of this inequality can be obtained by diagonalizing A and B simultaneously
and then using the following inequality which is a consequence of the arithmetic–
geometric mean inequality,
� (x1 + y1) ···(xd + yd) � 1 d ≥ (x1 ···xd) 1 d + (y1 ···yd) 1 d for xi, yi ≥ 0,
where equality holds if and only if x = (x1,...,xd), y = (y1,...,yd) are
linearly dependent.
See also Sect. 29.4.
Proof of Theorem 11.1. A simple compactness argument yields the existence of at
least one ellipsoid in C of maximum volume.
To see uniqueness, assume that, on the contrary, there are two distinct maximum
volume ellipsoids in C. Without loss of generality we may assume that their
volumes are equal to that of B d . By (1) these ellipsoids can be represented in the
form AB d , BB d with suitable d × d-matrices A, B ∈ P, where A �= B and
det A = det B = 1. Then 1 2 (A + B)Bd ⊆ C by the convexity of C. Since
det � 1 2 (A + B) � > 1 by (2), the ellipsoid 1 2 (A + B)Bd has greater volume than
the maximum volume ellipsoids AB d , BB d , a contradiction. ⊓⊔
Proof of Theorem 11.2. (i)⇒(ii) By (1), the family of all ellipsoids in C is represented
by the following set of symmetric positive definite matrices in P.
E = � A ∈ P : Au · v = A · u ⊗ v ≤ hC(v) for all u,v ∈ bd B d� .
Clearly, E is the intersection of the family of the closed halfspaces
(3) � A ∈ E 1 2 d(d+1) : A · u ⊗ v ≤ hC(v) � : u,v ∈ bd B d ,
O
K
H
P
D
u2 ⊗ u2
E I
I
u1 ⊗ u1
Fig. 11.1. Proof of John’s theorem