Gruber P. Convex and Discrete Geometry

228 Convex Bodies
The first such result in convex geometry seems to be due to Groemer [404]. A stability
result related to Blaschke’s characterization of ellipsoids was given by the author
[434].
Kakutani’s Characterization of Euclidean Norms
As a consequence of Blaschke’s characterization of ellipsoids, we will prove the
following characterization of Euclidean norms by Kakutani [559].
Theorem 12.5. Let |·|be a norm on E d , d ≥ 3, and k ∈{2,...,d − 1}. Then the
following statements are equivalent:
(i) |·|is Euclidean.
(ii) For each k-dimensional linear subspace S of E d there is a (linear) projection
pS : E d → S with norm |pS| =sup � |pS(x)| :x ∈ E d , |x| ≤1 � equal to 1.
The following proof was proposed by Klee [591]. It makes use of polarity. Here
polarity is defined slightly more generally than in Sect. 9.1.
C ∗ ={y : x · y ≤ 1fory ∈ C} for convex C ⊆ E d , o ∈ C.
The properties of polarity needed in the proof of Kakutani’s theorem and one further
property are collected together in Lemma 12.3, the proof of which is left to the
interested reader.
Lemma 12.3. The following properties hold:
(i) S linear subspace of E d ⇒ S ∗ = S ⊥
(ii) C ⊆ E d convex, o ∈ C ⇒ C ∗∗ = cl C
(iii) C, D ⊆ E d convex, o ∈ C ∩ D ⇒ (C ∩ D) ∗ = cl conv(C ∗ ∪ D ∗ )
(iv) E ⊆ E d ellipsoid with centre o ⇒ E ∗ ellipsoid with centre o
(v) C ⊆ E d convex, o ∈ C, S a linear subspace of E d ⇒ C + S = cl conv(C ∪ S)
Proof of the Theorem. It is easy to show that (i)⇒(ii).
(ii)⇒(i) By Lemma 12.1 it is sufficient to consider the case
d = k + 1, or k = d − 1.
Clearly, statement (ii) can be expressed as follows, where B ={x :|x| ≤1} is the
solid unit ball of the norm |·|.
Note that
(5) For each hyperplane S through o, there is a line T through o such that
B ∩ S = (B + T ) ∩ S.
B ∗ + S ∗ = cl conv(B ∗ ∪ S ∗ ) = (B ∩ S) ∗ = � (B + T ) ∩ S � ∗
= cl conv � (B + T ) ∗ ∪ S ∗� = (B + T ) ∗ + S ∗
= � cl conv(B ∪ T ) � ∗ + S ∗ = � cl conv(B ∗∗ ∪ T ∗∗ ) � ∗ + S ∗
= (B ∗ ∩ T ∗ ) ∗∗ + S ∗ = (B ∗ ∩ T ∗ ) + S ∗ ,