Let ${\mathcal A} \subseteq {\mathbb R}^n$ be closed and simply connected (also no inner "holes"). This set also has the following property: $\forall a \in {\mathcal A}$ and $\forall \varepsilon > 0$, ${\mathcal A} \bigcap \mu_n({\cal B}(a, \varepsilon)) > 0$, where ${\cal B}(a, \varepsilon)$ is an open ball, and $\mu_n(\cdot)$ returns the $n$-dimensional Lebesgue measure.
My question is: Is there any name for ${\mathcal A}$? Thanks!
PS: ${\mathcal A}$ can be $n$-dimensional ball, $n$-simplex, any convex set in ${\mathbb R}^n$ with $\mu_n({\cal A}) > 0$, etc. But ${\mathcal A}$ cannot be a Tootsie Pop (a $3$-dimensional ball with a $2$-dimensional line segement).