$D_r(x)$ denotes a closed ball of radius $r$ centered at $x$. Definition. Let $M \subset \mathbb{R}^n$. $D_r (M): = \bigcup\limits_{x \in M} D_r (x)$ $Int_r (M): = \{x ~|~ D_r(x) \subset M\}$ Question1: Is it true that $M \subset \mathbb{R}^n$ the f the following conditions are equivalent Ther...

Suppose that I'm given a $d$-dimensional closed and connected Riemannian manifold $(M,g)$. Then there exist $(K_n,\phi_n)_{n=1}^N$ such that $\phi_n:K_n \rightarrow \Delta_{d}$ is a homomorphism from some compact subset $K_n\subseteq M$ and the $d$-dimensional simplex: $ \Delta_d:=\{x\in [0,1]^{...