  We study the bottom of the spectrum in Hilbert geometries, we show that it is
zero if and only if the geometry is amenable, in other words if and only if it
admits a F\"olner sequence. We also show that the bottom of the spectrum admits
an upper bound, which depends only on the dimension and which is the bottom of
the spectrum of the Hyperbolic geometry of the same dimension. Horoballs, from
a purely metric point of view, and their relation with the bottom of the
spectrum in Hilbert geometries are briefly studied.
