Geometric and Functional Inequalities for Log-Concave Probability Sequences

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