Let $C\subseteq \mathbb R^n$ be non-empty, convex and compact. For $v\in S^{n-1}$, let $H_v$ be the supporting hyperplane in the direction of $v$ (i.e., $H_v$ is the boundary of the smallest closed half-space with outward normal $v$ that contains $C$). Let $U\subseteq S^{n-1}$ be the set of directions $v$ such that $H_v$ meets $C$ at exactly one point.

Main question: Does $S^{n-1}\backslash U$ have measure zero?

If not, then I have a second question: Is $U$ dense in $S^{n-1}$?

For $n=2$, it's easy to see that $S^{n-1}\backslash U$ is countable (otherwise there are uncountably many nondegenerate line segments in the boundary of $C$, and hence $C$ has infinite perimeter). But a cylinder shows that in general $S^{n-1}\backslash U$ need not be countable.