Gruber P. Convex and Discrete Geometry

12 Special Convex Bodies 229
by properties (v), (iii) of the lemma, Proposition (5) and, again, properties (iii), (v),
(ii), (iii) and (ii). Thus property (i) and Proposition (5) imply that
For each line L(= S ∗ ) through o there is a hyperplane H(= T ∗ ) through o
such that
B ∗ + L = B ∗ ∩ H + L.
Blaschke’s characterization of ellipsoids then shows that B ∗ is an ellipsoid. Since o is
in the centre of B, alsoB ∗ has centre o. An application of property (iv) then implies
that B is an ellipsoid, concluding the proof of Statement (i) of the theorem. ⊓⊔
Orthogonality in Normed Spaces
A well-known notion of orthogonality in normed spaces which goes back to a question
of Carathéodory on Finsler spaces is as follows. Given a normed space with
norm |·|, a vector x is orthogonal to a vector y, in symbols
x⊥y, if |x| ≤|x + λy| for all λ ∈ R.
Expressed geometrically, x⊥y means that the line through x parallel to the vector y
supports the ball {z :|z| ≤|x|} at the boundary point x. This notion of orthogonality
is sometimes named for Birkhoff [118] or James [541]. Orthogonality is important
in the context of approximation and still attracts interest, see the pertinent literature
in the books of Amir [27] and Istrǎtescu [537].
The following result is due to Blaschke [123, 124] and Birkhoff [118].
Theorem 12.6. Let |·|be a norm on E d , d ≥ 3, and let ⊥ be the corresponding
notion of orthogonality. Then the following are equivalent:
(i) |·|is Euclidean.
(ii) ⊥ is symmetric, i.e. x⊥y implies y⊥x forx, y ∈ E d .
Proof. (i)⇒(ii) This is trivial.
(ii)⇒(i) Let B ={x :|x| ≤1}. We will prove that
(6) For each line L through o there is a hyperplane H through o such that
B + L = B ∩ H + L.
Let bd B ∩ L ={±p}, say, and let H + p be a support hyperplane of B at p.Forthe
proof that B + L = B ∩ H + L, it is sufficient to show that any line of the form L +q,
where q ∈ B, meets B ∩ H. Given such a line, let (L + q) ∩ H ={r}. Since r ∈ H
and H + p supports B at p, wehave|p| ≤|p + λr| for all λ ∈ R or p⊥r. Thus
r⊥p by (ii), or |r| ≤|r + µp| for all µ ∈ R, or|r| ≤|s| for all s ∈ L + r = L + q.
Hence, in particular, |r| ≤|q| ≤1, or r ∈ B. This concludes the proof of (6).
Having proved (6), Blaschke’s ellipsoid theorem shows that B is an ellipsoid,
concluding the proof of (i). ⊓⊔