Log-Concave Functions And Poset Probabilities

elements of some (finite) poset , write for the probability that precedes in a random (uniform) linear extension of . For define

where the infimum is over all choices of and distinct .

Addressing an issue raised by Fishburn [6], we give the first nontrivial lower bounds on the function . This is part of a more general geometric result, the exact determination of the function

where the infimum is over chosen uniformly from some compact convex subset of a Euclidean space.

These results are mainly based on the Brunn–Minkowski Theorem and a theorem of Keith Ball [1], which allow us to reduce to a 2-dimensional version of the problem.