  Let $L$ be a lattice of finite length and let $d$ denote the minimum path
length metric on the covering graph of $L$. For any $\xi=(x_1,\dots,x_k)\in
L^k$, an element $y$ belonging to $L$ is called a median of $\xi$ if the sum
$d(y,x_1)+\cdots+d(y,x_k)$ is minimum. The lattice $L$ satisfies the
$c_1$-median property if, for any $\xi=(x_1,\dots,x_k)\in L^k$ and for any
median $y$ of $\xi$, $y\leq x_1\vee\dots\vee x_k$. Our main theorem asserts
that if $L$ is an upper semimodular lattice of finite length and the breadth of
$L$ is less than or equal to $2$, then $L$ satisfies the $c_1$-median property.
Also, we give a construction that yields semimodular lattices, and we use a
particular case of this construction to prove that our theorem is sharp in the
sense that $2$ cannot be replaced by $3$.
