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
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(thus fixing the curvature to be −4). Then the Carathéodory metric "d" on "B" is defined by
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What it means for a function on a Banach space to be holomorphic is defined in the article on Infinite dimensional holomorphy.
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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
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Moreover, any ℓ satisfying these three conditions has |ℓ("a" − "b")| = ||"a" − "b"||.
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"x""B" 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
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One can show that "α"("x", "v") ≥ ||"v"||, with equality when "x" = 0.