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 ∑ 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)=∨ 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