Long-concave functions and poset probabilities
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.
scientific article; zbMATH DE number 1336265