  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.
