Proof Mining - Mathematics, Algorithms and Proofs

Proof Mining
Hyperbolic Spaces
Definition 10 (Takahashi,Kirk,Reich)
A hyperbolic space is a triple (X, d, W ) where (X, d) is metric
space and W : X × X × [0, 1] → X s.t.
(i) d(z, W (x, y, λ)) ≤ (1 − λ)d(z, x) + λd(z, y),
(ii) d(W (x, y, λ), W (x, y, ˜ λ)) = |λ − ˜ λ| · d(x, y),
(iii) W (x, y, λ) = W (y, x, 1 − λ),
(iv) d(W (x, z, λ), W (y, w, λ)) ≤ (1 − λ)d(x, y) + λd(z, w).
Examples: Open unit disk D ⊂ C and Hilbert ball with
hyperbolic metric, Hadamard manifolds.
Ulrich Kohlenbach 12