**F. Aurenhammer**

A simple cell complex in Euclidean -space is a covering of
by finitely many convex -dimensional polyhedra (the -faces of ),
each of which is in the closure of exactly -faces of . An
algorithm that recognises when is the projection of the set of faces
bounding some convex polyhedron in , and that constructs
provided its existence is outlined. The method is optimal at least for
. No complexity results were previously known for both problems. The
results have applications in statics, to the recognition of Voronoi diagrams,
and to planar point location.