On the Banach-Mazur Ellipse

for details see e.g. [13,Section 6.1.1].

If X and Y are of one and the same finite dimension, then the minimum in (1) is attained.

An important characteristic of a finite dimensional space is its distance to the Euclidean space. Here we study this characteristic in the simplest case: the plane.

So, for a 2-dimensional normed space (X, ∥ • ∥)

where R 2 is considered with the standard Euclidean norm | • |. It is easy to check that

where B X and S X stand for the unit ball and the unit sphere of X respectively. Indeed, T (B X ) ⊃ B R 2 is equivalent to ∥T -1 ∥ ≤ 1, so we have that d 2 (X) is at most the right-hand side of (2). On the other hand, multiplying the operator T by a real constant does not change ∥T ∥ • ∥T -1 ∥, so we may assume that ∥T -1 ∥ = 1 in (1).

Using polar decomposition, we can reduce (2). Note that if U is in O(2) -the group of isometries of R 2 , that is, U -1 = U t if considered as a matrix -then ∥U T ∥ = max x∈S X |U T x| = max x∈S X |T x| = ∥T ∥, and similarly ∥T -1 U -1 ∥ = ∥T -1 ∥, so we can quotient by O(2). Denote by PD(X, R 2 ) the operators from X to R 2 that are positive definite symmetric matrices in any basis. By [4,Theorem 7.3.1,p.449] each isomorphism T : X → R 2 can be represented as T = U T 1 , where U ∈ O(2) and T 1 ∈ PD(X, R 2 ). Therefore,

We prove in a new way the following characterisation, which is essentially contained in [1,10].

There are x i ∈ S X , i = 1, 2, with x 1 ̸ = ±x 2 , and

Here cone {x, y} := {αx + βy : α, β ≥ 0}.

Note that the condition characterising the optimal operator depends only on the operator itself and not on d 2 (X). This is to be expected, because the problem suitably stated is convex. Not strictly convex, however, so the uniqueness is not immediate. As an illustration, consider the following picture.

T

From it we can also see one important, but immediate, corollary. Since one of the angles between T y 1 and T y 2 , and between T y 2 and -T y 1 (say the one between T y 1 and T y 2 as on the picture) will be ≤ π/2; and since T x 2 will be within the angle between the tangents to the unit circle at T y 1 and T y 2 , due to convexity, the distance from T x 2 to zero, that is

Ader’s Non-separation Principle ( [1]). For the unique T there is no angle such that all points x ∈ S X with | T x| = ∥ T ∥ are inside this angle and all points y ∈ S X with | T y| = 1 are outside.

Here, an angle is cone {a, b} ∪ cone {-a, -b} for some a, b ∈ R 2 . The original work [1] states that there should be at most 4 points realising Ader’s Nonseparation Principle, and for these 4 points it is then clear that the alternance described in Theorem 1 must be in place. By Maurey’s result [10], see also [3], the ellipse of minimal Banach-Mazur distance is unique, which yields the uniqueness part of Theorem 1.

Obviously, the problem can be re-stated more geometrically like: find the minimal k ≥ 1 such that there is an ellipse inscribed in B X whose homothetic image with ratio k is circumscribed about B X . We will actually be using this form below, but we mention it here to point out the relationship to the celebrated Löwner-John ellipse, see e.g. [7], that is, the ellipse of maximal area inscribed in B X . The Löwner-John ellipsoid is extensively used for estimating the Banach-Mazur distance to Euclidean space, probably because Ader’s result [1] had apparently been forgotten until Grundbacher and Kobos recently unearthed it, see [3].

However, if B X is not symmetric the estimate through the Löwner-John ellipsoid can grow progressively worse with the dimension. Consider R n and let B Xn be the convex hull of B R n and √ ne, where e ∈ S R n . Then B R n is the Löwner-John ellipsoid for B Xn , see [14] for a nice and simple proof, so it gives only the generic estimate √ n. On the other hand, by the rotational symmetry of B Xn around the axis through e, d(X n , R n ) ≤ √ 3. Our interest in the specifically two-dimensional case comes from our previous works [6,5,11,12]. Of course, similar problems can also be considered in different metrics, like e.g. the Hausdorff, see [8,2,9].

In conclusion, note that the convexity of B X will not be used at all in the proofs below, which is not surprising, as they go through Chebyshev approximation.

In view of (3), Theorem 1 characterises the solution of the optimisation problem

By substituting T -1 in place of T , this problem can be rewritten as

which is more suited for our purposes. Theorem 1 is immediately equivalent to the following Proposition 1 with which we will be working from now on.

Proposition 1. The only solution to (8), say T , is characterised by the following property:

Let a co-ordinate system be fixed in X in which

for a strictly positive, π-periodic and continuous r : R → R + . In these fixed coordinates each T ∈ PD(R 2 , X) is identified with a matrix

Obviously,

That