  Topologically stable cellular partitions of D dimensional spaces are studied.
A complete statistical description of the average structural properties of such
partition is given in term of a sequence of D/2-1 (or (D-1)/2) variables for D
even (or odd). These variables are the average coordination numbers of the
2k-dimensional polytopes (2k < D) which make the cellular structure. A
procedure to built D dimensional space partitions trough cell-division and
cell-coalescence transformations is presented. Classes of structures which are
invariant under these transformations are found and the average properties of
such structures are illustrated. Homogeneous partitions are constructed and
compared with the known structures obtained by Voronoi partitions and sphere
packings in high dimensions.
