Geometric and Functional Inequalities for Log-concave Probability Sequences

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.