A sub-Riemannian Santaló formula with applications to isoperimetric inequalities and first Dirichlet eigenvalue of hypoelliptic operators (Q1717637)

The Santaló formula [\textit{I. Chavel}, Riemannian geometry. A modern introduction. 2nd ed. Cambridge: Cambridge University Press (2006; Zbl 1099.53001); \textit{L. Santaló}, Integral geometry and geometric probability. With a foreword by Mark Kac. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1116.53050)] describes the intrinsic Liouville measure on the unit cotangent bundle in terms of the geodesic flow. \par Thus, if $(M,g)$ is a compact connected Riemannian manifold with boundary $ \partial M$, $\mu $ the Liouville measure of the unit tangent bundle $UM$ in terms of the geodesic flow $\phi _{t}:UM\rightarrow UM$, then for any measurable function $F:UM\rightarrow\mathbb{R}$ we have \[ \int_{U^{\Diamond }M}F\mu =\int_{\partial M}\left[ \int_{U_{q}^{+}\partial M}\left(\int_{0}^{l(v)}F(\phi _{t}(v))dt\right) g(v,\mathbf{n}_{q})\mathbf{ \eta }_{q}(v)\right] \sigma (q), \] where $\sigma $ is the surface form on $\partial M$ induced by the inward pointing normal vector $\mathbf{n}$, $\eta _{q}$ is the Riemannian spherical measure on $U_{q}M$, $U_{q}^{+}\partial M$ is the set of inward pointing unit vectors at $q\in \partial M$, $l(v)$ is the exit length of the geodesic with initial vector $v$ and $U^{\diamond}M\subseteq UM$ is the visible set (the set of unit vectors that can be reached via the geodesic flow starting from points on $\partial M$). \par In the paper under review, the authors prove the reduced Santaló formula and apply it to prove Poincaré, Hardy and isoperimetric-type inequalities.