Uniform estimation of sub-Riemannian balls

A fundamental result of sub-Riemannian geometry, the ball-box theorem, states that small sub-Riemannian balls look like boxes [-ε^w₁, ε^w₁]X ⋯ X[-ε^w_n, ε^w_n] in privileged coordinates. This description is not uniform in general. Thus, it does allow us neither to compute Hausdorff measures and dimensions nor to prove the convergence of certain motion planning algorithms. In this paper, we present a description of the shape of small sub-Riemannian balls depending uniformly on their center and their radius. This result is a generalization of the ball-box theorem. The proof is based on the one hand on a lifting method, which replaces the sub-Riemannian manifold by an extended equiregular one (where the ball-box theorem is uniform); and on the other hand, it based on an estimate of sets defined by families of vector fields, which allows us to project the balls in suitable coordinates.