As the question suggests, does the fixed point in the Brouwer's Fixed-Point Theorem have to be an interior point? Many thanks in advance.
To be clear, I'm using the statement of Brouwer's Fixed-Point Theorem from $\S$2.2 of Guillemin-Pollack, which is stated as follows.
Any smooth map $f$ of the closed unit ball $B^n \subset \mathbb{R}^n$ into itself must have a fixed point; that is, $f(x) = x$ for some $x \in B^n$.
No. Map $[0,1]$ to itself by division by $2$. Or the same for any convex compact set in Euclidean space, take a boundary point and shrink the whole set toward it by a factor of $1/2$.