Uniform estimation of sub-Riemannian balls. (English) Zbl 1029.53039

In the sub-Riemannian geometry, the shape of small balls is described by the so-called ball-box theorem. But, as it is well-known, the ball-box theorem does not give a uniform description for the small sub-Riemannian balls covering a sub-Riemannian manifold. Consequently, it does not allow us either to compute Hausdorff measures and dimensions or to prove the convergence of certain motion planning algorithms. The author succeeded in proving a generalization of the box-ball theorem. More exactly, he gives a description of the shape of small sub-Riemannian balls depending uniformly on their center and their radius. The proof is based, on the one hand, on a lifting method which replaces the sub-Riemannian manifold by an extended equiregular one (with no singular points), where the box-ball theorem is uniform; and, on the other hand, it is based on an estimate of sets defined by families of vector fields, which allows him to project the balls in suitable coordinates.


53C17 Sub-Riemannian geometry
93B29 Differential-geometric methods in systems theory (MSC2000)
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