  We build polyhedral complexes in Rn that coincide with dyadic grids with
different orientations, while keeping uniform lower bounds (depending only on
n) on the flatness of the added polyhedrons including their subfaces in all
dimensions. After the definitions and first properties of compact Euclidean
polyhedrons and complexes, we introduce a tool allowing us to fill with
n-dimensionnal polyhedrons a tubular-shaped open set, the boundary of which is
a given n - 1-dimensionnal complex. The main result is proven inductively over
n by completing our dyadic grids layer after layer, filling the tube
surrounding each layer and using the result in the previous dimension to build
the missing parts of the tube boundary. A possible application of this result
is a way to find solutions to problems of measure minimization over certain
topological classes of sets, in arbitrary dimension and codimension.
