2 research outputs found
Hilbert geometry of the Siegel disk: The Siegel-Klein disk model
We study the Hilbert geometry induced by the Siegel disk domain, an open
bounded convex set of complex square matrices of operator norm strictly less
than one. This Hilbert geometry yields a generalization of the Klein disk model
of hyperbolic geometry, henceforth called the Siegel-Klein disk model to
differentiate it with the classical Siegel upper plane and disk domains. In the
Siegel-Klein disk, geodesics are by construction always unique and Euclidean
straight, allowing one to design efficient geometric algorithms and
data-structures from computational geometry. For example, we show how to
approximate the smallest enclosing ball of a set of complex square matrices in
the Siegel disk domains: We compare two generalizations of the iterative
core-set algorithm of Badoiu and Clarkson (BC) in the Siegel-Poincar\'e disk
and in the Siegel-Klein disk: We demonstrate that geometric computing in the
Siegel-Klein disk allows one (i) to bypass the time-costly recentering
operations to the disk origin required at each iteration of the BC algorithm in
the Siegel-Poincar\'e disk model, and (ii) to approximate fast and numerically
the Siegel-Klein distance with guaranteed lower and upper bounds derived from
nested Hilbert geometries.Comment: 42 pages, 7 figure