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Filling Euclidean space using polyhedral complexes of given orientation and uniform rotundity. (Remplissage de l’espace euclidien par des complexes polyédriques d’orientation imposée et de rotondité uniforme.) (French. English summary) Zbl 1261.52009
Author’s abstract: We build polyhedral complexes in $$\mathbb R^n$$ 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 $$n$$-dimensional polyhedrons a tubular-shaped open set, the boundary of which is a given $$n-1$$-dimensional 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.

##### MSC:
 52B70 Polyhedral manifolds
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