Let $(M^n,g)$ be a smooth complete Riemannian manifold. Let $p\in M$ be a point. Recall that the cut locus of $p$ is the set of vectors $v$ in the tangent space $T\_pM$ such that $\exp(t v)$ is a minimizing geodesic for any $t\in [0,1]$, but not for $t\in [0,1+\varepsilon )$ for any $\varepsilon >0$.


**Question.** Does the cut locus have Lebesgue measure 0 in $T\_pM$? If yes, does it have Hausdorff dimension at most $n-1$?


If the above questions have negative answers, one may ask the same questions about the exponential image of the cut locus.



Geometry of the cut locus
The key fact is that the cut time $t\_c : UM \to \mathbb{R}$, defined on the unit tangent bundle $UM$ of a complete, $n$-dimensional Riemannian manifold, is locally Lipschitz continuous around all $v \in UM$ such that $t\_c(v) < +\infty$. Hence the tangential cut locus at $p \in M$, that is 
$$
\tilde{C}\_p = \{t\_c(v)v\mid v \in UM,\quad t\_c(v) < +\infty\} \subset T\_q M,
$$
either is empty, or it has Hausdorff dimension exactly $n-1$ (being the graph of a locally Lipschitz function). The exponential map being smooth, it cannot increase the Hausdorff dimension, hence $\dim(C\_p) = \dim(\exp\_p(\tilde{C}\_p)) \leq n-1$.


All of this is proved [here](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.309.211&rep=rep1&type=pdf) by Itoh and Tanaka.


