We investigate various 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 analog 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. This amounts to a discrete analog of the localization lemma of Lovasz and Simonovits. Further applications of this lemma are used to produce a discrete version of the Prekopa-Leindler inequality, large deviation inequalities for log-concave measures about their mean, and provide insight on the stability of generalized log-concavity under convolution.
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