  The tight span $T_d$ of a metric $d$ on a finite set is the subcomplex of
bounded faces of an unbounded polyhedron defined by~$d$. If $d$ is generic then
$T_d$ is known to be dual to a regular triangulation of a second hypersimplex.
A tight upper and a partial lower bound for the face numbers of $T_d$ (or the
dual regular triangulation) are presented.
