  Reflexive polyhedra encode the combinatorial data for mirror pairs of
Calabi-Yau hypersurfaces in toric varieties. We investigate the geometrical
structures of circumscribed polytopes with a minimal number of facets and of
inscribed polytopes with a minimal number of vertices. These objects, which
constrain reflexive pairs of polyhedra from the interior and the exterior, can
be described in terms of certain non-negative integral matrices. A major tool
in the classification of these matrices is the existence of a pair of weight
systems, indicating a relation to weighted projective spaces. This is the
corner stone for an algorithm for the construction of all dual pairs of
reflexive polyhedra that we expect to be efficient enough for an enumerative
classification in up to 4 dimensions, which is the relevant case for Calabi-Yau
compactifications in string theory.
