  We investigate geometric and functional inequalities for the class of
log-concave probability sequences. We prove dilation inequalities for
log-concave probability measures on the integers. A functional analogue of this
geometric inequality is derived, giving large and small deviation inequalities
from a median, in terms of a modulus of regularity. Our methods are of
independent interest, we find that log-affine sequences are the extreme points
of the set of log-concave sequences belonging to a half-space slice of the
simplex. We use this result as a tool to derive simple proofs of several
convolution type inequalities for log-concave sequences, due to Walkup,
Gurvits, and Klartag-Lehec. Further applications of our results are used to
produce a discrete version of the Pr\'ekopa-Leindler inequality.
