, Volume 18, Issue 1, pp 85-99

Log-Concave Functions And Poset Probabilities

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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.
Received: October 6, 1997