Carathéodory metric

In mathematics, the Carathéodory metric is a metric defined on the open unit ball of a complex Banach space that has many similar properties to the Poincaré metric of hyperbolic geometry. It is named after the Greek mathematician Constantin Carathéodory.

Definition

Let (X, || ||) be a complex Banach space and let B be the open unit ball in X. Let Δ denote the open unit disc in the complex plane C, thought of as the Poincaré disc model for 2-dimensional real/1-dimensional complex hyperbolic geometry. Let the Poincaré metric ρ on Δ be given by

[math]\displaystyle{ \rho (a, b) = \tanh^{-1} \frac{| a - b |}{|1 - \bar{a} b |} }[/math]

(thus fixing the curvature to be −4). Then the Carathéodory metric d on B is defined by

[math]\displaystyle{ d (x, y) = \sup \{ \rho (f(x), f(y)) | f : B \to \Delta \mbox{ is holomorphic} \}. }[/math]

What it means for a function on a Banach space to be holomorphic is defined in the article on Infinite dimensional holomorphy.

Properties

• For any point x in B,

[math]\displaystyle{ d(0, x) = \rho(0, \| x \|). }[/math]

• d can also be given by the following formula, which Carathéodory attributed to Erhard Schmidt:

[math]\displaystyle{ d(x, y) = \sup \left\{ \left. 2 \tanh^{-1} \left\| \frac{f(x) - f(y)}{2} \right\| \right| f : B \to \Delta \mbox{ is holomorphic} \right\} }[/math]

• For all a and b in B,

[math]\displaystyle{ \| a - b \| \leq 2 \tanh \frac{d(a, b)}{2}, \qquad \qquad (1) }[/math]

with equality if and only if either a = b or there exists a bounded linear functional ℓ ∈ X^∗ such that ||ℓ|| = 1, ℓ(a + b) = 0 and

[math]\displaystyle{ \rho (\ell (a), \ell (b)) = d(a, b). }[/math]

Moreover, any ℓ satisfying these three conditions has |ℓ(a − b)| = ||a − b||.

• Also, there is equality in (1) if ||a|| = ||b|| and ||a − b|| = ||a|| + ||b||. One way to do this is to take b = −a.
• If there exists a unit vector u in X that is not an extreme point of the closed unit ball in X, then there exist points a and b in B such that there is equality in (1) but b ≠ ±a.

Carathéodory length of a tangent vector

There is an associated notion of Carathéodory length for tangent vectors to the ball B. Let x be a point of B and let v be a tangent vector to B at x; since B is the open unit ball in the vector space X, the tangent space T_xB can be identified with X in a natural way, and v can be thought of as an element of X. Then the Carathéodory length of v at x, denoted α(x, v), is defined by

[math]\displaystyle{ \alpha (x, v) = \sup \big\{ | \mathrm{D} f(x) v | \big| f : B \to \Delta \mbox{ is holomorphic} \big\}. }[/math]

One can show that α(x, v) ≥ ||v||, with equality when x = 0.

See also

• Earle–Hamilton fixed point theorem

References

• Earle, Clifford J. and Harris, Lawrence A. and Hubbard, John H. and Mitra, Sudeb (2003). "Schwarz's lemma and the Kobayashi and Carathéodory pseudometrics on complex Banach manifolds". Kleinian groups and hyperbolic 3-manifolds (Warwick, 2001). London Math. Soc. Lecture Note Ser. 299. Cambridge: Cambridge Univ. Press. pp. 363–384. ISBN 9780521540131. https://archive.org/details/kleiniangroupshy00komo.