I am new to topology and getting confused about some notions.
The question is simple. Are these two statements are equivalent?:
- $M$ is a convex smooth $p$-manifold with boundary.
- $M$ is a convex and closed subset of $\mathbb{R}^p$ whose boundary is a smooth orientable surface.
Thank you!
Examples are closed $p$-ball $\{x \in \mathbb{R}^p | \Vert x \Vert_{2} \leq 1\} $ and $\{x \in \mathbb{R}^3 | x_{2} \ge x_{1}^2\}$.