Minkowski content for reachable sets

In 1955, Martin Kneser showed that the Minkowski content of a compact p-rectifiable subset M of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}^n}$$\end{document} is equal to its p-Hausdorff measure: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim_{t\to 0, t > 0}\frac{{\mathcal{L}}^n\left(\overline{B}(M,t)\right)}{\alpha(n-p) t^{(n-p)}}={\mathcal{H}}^p(M).$$\end{document}We extend his result to the reachable sets of a linear control system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{x}= f(x) u,$$\end{document}and we give an interpretation in terms of a Riemannian distance.