One can prove that a differentiable function $f\colon \mathbb{R}\to \mathbb{R}$ is strictly convex if and only if for each $a\in\mathbb{R}$ we have $$f'(a)(x-a)+f(a)\leq f(x)\qquad\text{for all }x\in\mathbb{R},$$ where equality occurs only at $x=a$. If this is the case, then the tangent line of $f$ through $(a,f(a))$ is the unique line through $(a,f(a))$ which has no intersection with the set $\{(x,y)\in\mathbb{R}^2:y>f(x)\}. $ In a similar vein, is the following statement true?
Let $M\subset\mathbb{R}^n$ be a smooth, compact, connected, $n$-dimensional submanifold-with-boundary. Then $M$ is a convex set if and only if for each $x\in \partial M$, the tangent space $(\partial M)_x$ has no intersection with $M\setminus\partial M$. If this is the case, then $(\partial M)_x$ is the unique $(n-1)$-dimensional hyperplane through $x$ which has no intersection with $M\setminus\partial M$.