Let $X^n$ be an $n$-dimensional complete Alexandrov space with curvature bounded below (or a smooth Riemannian manifold, possibly with boundary). Let $U\subset X$ be an open dense subset with the following convexity property: Any shortest path connecting any two points of $U$ is contained in $U$.
Question 1. Is the Hausdorff dimension of $X\backslash U$ at most $n-1$?
Question 2. If the answer to question 1 is yes, whether the corresponding Hausdorff measure of $X\backslash U$ is locally finite?
The answer is not known to me even in the case of Riemannian manifolds.
A reference would be helpful.