  We begin a coarse geometric study of Hilbert geometry. Actually we give a
necessary and sufficient condition for the natural boundary of a Hilbert
geometry to be a corona, which is a nice boundary in coarse geometry. In
addition, we show that any Hilbert geometry is uniformly contractible and with
coarse bounded geometry. As a consequence of these we see that the coarse
Novikov conjecture holds for a Hilbert geometry with a mild condition. Also we
show that the asymptotic dimension of any two-dimensional Hilbert geometry is
just two. This implies that the coarse Baum-Connes conjecture holds for any
two-dimensional Hilbert geometry via Yu's theorem.
