Gruber P. Convex and Discrete Geometry

206 Convex Bodies
with P. Thus, it is the intersection of a closed convex set with P and therefore convex.
By (i), det A < 1 for all A ∈ E \{I }. This, together with (2), yields the following:
(4) E is convex, D ∩ E ={I } and E is separated from D by the unique support
hyperplane H of D at I .
The support cone K of E at I is the closed convex cone with apex I generated
by E. Since E is the intersection of the closed halfspaces in (3) and P, and since
these halfspaces vary continuously as u,v range over bd B d , the support cone K is
the intersection of those halfspaces in (3) which contain the apex I of K on their
boundaries, i.e. I · u ⊗ v = u · v = hC(v). Since u · v ≤ 1, hC(v) ≥ 1foru,v ∈
bd B d , the equality I · u ⊗ v = hC(v) holds precisely in case where u = v and
v ∈ bd B d ∩ bd C. Thus K is the intersection of the halfspaces
� A ∈ E 1 2 d(d+1) : A · u ⊗ u ≤ 1 � : u ∈ bd B d ∩ bd C = B d ∩ bd C.
The normal cone N, ofE at I , is the polar cone of K − I and thus is generated by
the exterior normals u ⊗ u, u ∈ B d ∩ bd C, of these halfspaces,
(5) N = pos � u ⊗ u : u ∈ B d ∩ bd C � .
The cone K has apex I and, by (4), is separated form D by the hyperplane H.
The normal I of H points away from K and thus is contained in the normal cone N.
Noting (5), Carathéodory’s theorem then implies that there are ui ∈ Bd ∩ bd C,
λi > 0, i = 1,...,m, where m ≤ 1 2d(d + 1), such that
I = �
This, in turn, shows that
i
λi ui ⊗ ui.
d = trace I = �
λi trace ui ⊗ ui = �
λi.
i
For the proof that m ≥ d, it is sufficient to show that lin{u1,...,um} =E d .Ifthis
were not the case, we could choose a unit vector u orthogonal to u1,...,um to obtain
the contradiction
1 = u 2 = Iu · u = �
i
λi
� (ui ⊗ ui) u � · u = �
i
λi
i
� �
(ui · u) ui · u = 0.
(ii)⇒(i) Let E be as above. E is convex and I is a boundary point of it by (ii).
Thus we may define K, N as before. (ii) yields I ∈ N. The hyperplane H through I
and orthogonal to I thus separates K and D and thus, a fortiori, D and E. Since D is
strictly convex by (2), D ∩ E ={I }. This shows that det A < 1 for each A ∈ E \{I },
or, in other words, B d is the unique ellipsoid in C with maximum volume. ⊓⊔
Remark. For the proof of a more general implication (i)⇒(ii), see Giannopoulos,
Perissinaki and Tsolomitis [376]. Proofs of John’s theorem (see Fig. 11.1) in the
non-symmetric case and of the generalized version of Giannopoulos, Perissinaki and
Tsolomitis in the spirit of the above proof are outlined in Gruber and Schuster [452].