Dynamical partitions of space in any dimension

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.