  We discuss the computational complexity of special cases of the 3-dimensional
(axial) assignment problem where the elements are points in a Cartesian space
and where the cost coefficients are the perimeters of the corresponding
triangles measured according to a certain norm. (All our results also carry
over to the corresponding special cases of the 3-dimensional matching problem.)
  The minimization version is NP-hard for every norm, even if the underlying
Cartesian space is 2-dimensional. The maximization version is polynomially
solvable, if the dimension of the Cartesian space is fixed and if the
considered norm has a polyhedral unit ball. If the dimension of the Cartesian
space is part of the input, the maximization version is NP-hard for every $L_p$
norm; in particular the problem is NP-hard for the Manhattan norm $L_1$ and the
Maximum norm $L_{\infty}$ which both have polyhedral unit balls.
