Given a compact set $E$ with non-empty interior in $R^d$ and some small positive number $r$, what kind of conditions should we put on the set $E$ so that for all $x\in E$, the volume of the intersection of $B(x,r)$ with $E$ is uniformly bounded away from $0$? Here $B(x,r)$ is the ball of radius $r$ centered at $x$.



conditions on the boundary of a compact set to ensure the volume of the intersection of a small ball with the set doesn't vanish
For any fixed $r$, $f(x):=vol(B(x,r)\cap E)$ is Lipschitz, hence continuous, with respect to $x$. Indeed, $\vert f(x)-f(y)\vert$ is at most the volume of the symmetrical difference of the balls. Hence, the minimum of $f$ is reached at some $x\in E$. So, it is enough that $f>0$. This will be the case if you suppose e.g. that $E$ is the closure of its interior, a hypothesis which looks natural in your case. For example, $\partial E$ being a $C^0$ ($d-1$)-dimensional submanifold of $R^d$ is enough.


