Sharp comparison of moments and the log-concave moment problem

This article investigates sharp comparison of moments for various classes of random variables appearing in a geometric context. In the first part of our work we find the optimal constants in the Khintchine inequality for random vectors uniformly distributed on the unit ball of the space ℓqn for q∈(2,∞), complementing past works that treated q∈(0,2]∪{∞}. As a byproduct of this result, we prove an extremal property for weighted sums of symmetric uniform distributions among all symmetric unimodal distributions. In the second part we provide a one-to-one correspondence between vectors of moments of symmetric log-concave functions and two simple classes of piecewise log-affine functions. These functions are shown to be the unique extremisers of the p-th moment functional, under the constraint of a finite number of other moments being fixed, which is a refinement of the description of extremisers provided by the generalised localisation theorem of Fradelizi and Guédon (2006) [7].