Singular Points and an Upper Bound of Medians in Upper Semimodular Lattices

Given a metric space (X, d) and a k-tuple (profile) P = (x1, x2,...,xk ) of elements of P X, a median is an element m of X minimizing the remoteness r(m) = i d(m, xi). In a special case that X is a finite upper semimodular lattice, Leclerc proved [9] that a lower bound of the medians is c(P ), the majority rule. By duality, an upper bound of the medians is c0 (P ), the dual majority rule, if X is a lower semimodular lattice. Both c(P ) and c0 (P ) will be defined in Section 2. These imply that in a finite modular lattice, the upper and the lower bounds of the medians will be c0 (P ) and c(P ), respectively. In the same paper, Leclerc pointed out an open problem: what is an upper bound for the medians in an upper semimodular lattice? Is c1(P ) (= W i xi) the upper bound of the medians? This paper shows that the upper bound is not c1(P ) by some examples (in Section 3) and gives the best possible upper bound (in Section 4).