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.
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
\rho(a,b)=\tanh-1
| |a-b| | |
| |1-\bar{a |
b|}
(thus fixing the curvature to be -4). Then the Carathéodory metric d on B is defined by
d(x,y)=\sup\{\rho(f(x),f(y))|f:B\to\Deltaisholomorphic\}.
What it means for a function on a Banach space to be holomorphic is defined in the article on Infinite dimensional holomorphy.
d(0,x)=\rho(0,\|x\|).
d(x,y)=\sup\left\{\left.2\tanh-1\left\|
| f(x)-f(y) | |
| 2 |
\right\|\right|f:B\to\Deltaisholomorphic\right\}
\|a-b\|\leq2\tanh
| d(a,b) | |
| 2 |
, (1)
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
\rho(\ell(a),\ell(b))=d(a,b).
Moreover, any ℓ satisfying these three conditions has |ℓ(a - b)| = ||a - b||.
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 TxB 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
\alpha(x,v)=\sup\{|Df(x)v||f:B\to\Deltaisholomorphic\}.
One can show that α(x, v) ≥ ||v||, with equality when x = 0.