Comparing moments of real log-concave random variables.

Log-concave random variables play an important role in probability theory. Moment comparison inequalities of the form ∥X∥_p ≤ C_{p,q}∥X∥_q are particularly useful in concentration of measure and convex geometry. In this talk, I will present optimal bounds of the form ∥X∥p  ≤   C_{p, q} ∥X∥q for real log-concave random variables and show that for any p > q > 0 the maximum of the ratio ∥X∥_p /∥X∥_q is attained for some shifted exponential distribution.