A note on the diameter of small sub-Riemannian balls

2025/05/05 by Di Marco, Marco, Somma, Gianluca, Vittone, Davide
#28A75 #53C17 #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2505.02790

Abstract

We observe that the diameter of small (in a locally uniform sense) balls in C^1,1 sub-Riemannian manifolds equals twice the radius. We also prove that, when the regularity of the structure is further lowered to C⁰, the diameter is arbitrarily close to twice the radius. Both results hold independently of the bracket-generating condition.

Citations

• Surface measure on, and the local geometry of, sub-Riemannian manifolds
• A Hamiltonian approach to sufficiency in optimal control with minimal regularity conditions: Part I
• Shortest paths for sub-Riemannian metrics on rank-two distributions

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