LOG-CONCAVE FUNCTIONS AND POSET PROBABILITIES

For a, y elements of some (finite) poset P, write p(x < y) for the pro bability that a precedes y in a random (uniform) linear extension of P . For u,v is an element of [0,1] define delta(u,v) = inf{p(x < z) : p( x < y) greater than or equal to u, p(y < z) greater than or equal to v }, where the infimum is over all choices of P and distinct x,y,z is an element of P. Addressing an issue raised by Fishburn [6], we give the first nontrivial lower bounds on the function 6. This is part of a mo re general geometric result, the exact determination of the function g amma(u, v) = inf(Pr(X-1 < X-3) :Pr(X-1 < X-2) greater than or equal to u, Pr(X-2 < X-3) greater than or equal to v), where the infimum is ov er X = (X-1,...,X-n) chosen uniformly from some compact convex subset of a Euclidean space. These results are mainly based on the Brunn-Mink owski Theorem and a theorem of Keith Ball [1], which allow us to reduc e to a a-dimensional version of the problem.