Long-concave functions and poset probabilities
type: Item
instance of: scholarly article
title: Long-concave functions and poset probabilities
zbMATH Open document ID: 0928.52006
DOI: 10.1007/PL00009812
author: Yu Yang
author: J. Kahn
published in: Combinatorica
publication date: 1999-09-14T00:00:00Z
review text: For \(x\), \(y\) elements of some (finite) partially ordered set (poset) \(P\), write \(p(x,y)\) for the probability that \(x\) preceeds \(y\) in a random (uniform) linear extension of \(P\). For \(u,v\in [0,1]\) define \[ \delta (u,v)=\inf \{p(x<z):p(x<y)\geq u, \;p(y<z)\geq v\}, \] where the infimum is over all choices of \(P\) and distinct \(x,y,z\in P\). Addressing an issue raised by Fishburn, the authors give the first nontrivial lower bounds on the function \(\delta\). This is part of a more general geometric result, the exact determination of the function \[ \gamma(u,v)=\inf\{Pr(X_1<X_3):Pr(X_1<X_2)\geq u,\;Pr(X_2<X_3)\geq v\}, \] where the infimum is over \(X=(X_1,\dots , X_n\}\) 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 allows to reduce to a 2-dimensional version of the problem.
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A41
Mathematics Subject Classification ID: 06A07
zbMATH DE Number: 1336265
zbMATH Keywords: poset probabilities
zbMATH Keywords: long-concave functions
reviewed by: Elisabeth M. Werner
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1336265