We present several functional inequalities
for finite difference gradients, such as
a Cheeger inequality, Poincaré and (modified) logarithmic Sobolev inequalities,
associated deviation estimates,
and an exponential integrability property.
In the particular case of the geometric distribution on ${\mathbb{N}}$
we use an integration by parts formula to compute
the optimal isoperimetric and Poincaré constants,
and to obtain an improvement of our
general logarithmic Sobolev inequality.
By a limiting procedure we recover the corresponding
inequalities for the exponential distribution.
These results have applications to interacting spin systems under
a geometric reference measure.