Singular points and an upper bound of medians in upper semimodular lattices

Given a k-tuple P=(x(1),x(2),...,x(k)) in a finite lattice X endowed with the lattice metric d, a median of P is an element m of X minimizing the sum Sigma (i)d(m,x(i)). If X is an upper semimodular lattice, Leclerc proved that a lower bound of the medians is c(P), the majority rule and he pointed out an open problem: "Is c(1)(P)=boolean OR (i)x(i), the upper bound of the medians?" This paper shows that the upper bound is not c(1)(P) and gives the best possible upper bound.