Remplissage De L'Espace Euclidien Par Des Complexes Poly\'Edriques D'Orientation Impos\'Ee Et De Rotondit\'E Uniforme

TL;DR

This paper introduces a method to construct uniform, oriented polyhedral complexes in Euclidean space that fill tubular regions around complex boundaries, aiding in measure minimization problems across dimensions.

Contribution

It presents a novel inductive construction of polyhedral complexes that fill tubular regions with uniform flatness bounds, applicable in arbitrary dimensions.

Findings

Constructed polyhedral complexes with uniform flatness bounds

Developed an inductive filling method for tubular regions

Potential applications in measure minimization problems

Abstract

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.