  We introduce the non-pure versions of simplicial balls and spheres with
minimum number of vertices. These are a special type of non-homogeneous balls
and spheres (NH-balls and NH-spheres) satisfying a minimality condition on the
number of maximal simplices. The main result is that minimal NH-balls and
NH-spheres are precisely the simplicial complexes whose iterated Alexander
duals converge respectively to a simplex or the boundary of a simplex.
