Let $(M^n,g)$ be a complete, $n$-dimensional Riemanian manifold without boundary, maybe non-compact. Let $p\in M$ be a point, and $C_p$ the cut locus. It's known that $C_p$ has Hausdorff dimension $\le n-1$. $C_p$ may not be bounded; for example, consider the cylinder. For $R>0$, consider the subset $$ C_{p,R}=C_p\cap \overline{B(p,R)}. $$ Let $A_\delta$ be the $\delta$-neighborhood of $C_{p,R}$, i.e. $$ A_\delta=\{x\in M, d(x,C_{p,R})<\delta\}. $$ My question is: For any $\epsilon>0$, is there $\delta>0$, such that $$ H^n(A_\delta)<\epsilon? $$