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.