Log-Concave Functions And Poset Probabilities
- Cite this article as:
- Kahn, J. & Yu, Y. Combinatorica (1998) 18: 85. doi:10.1007/PL00009812
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 , 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 , which allow us to reduce to a 2-dimensional version of the problem.
AMS Subject Classification (1991) Classes: 52A40, 52C07, 06A07
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© János Bolyai Mathematical Society, 1998