Let $B^-$ be the open unit ball in $\mathbb{R}^n$ for $n\in\mathbb{N}$, and let $f$ be a uniformly continuous function from $B^-$ to itself, with a uniformly continuous inverse $f^{-1}$.


Under these conditions, must $f$ have a fixed point in $B^-$? 



Brouwer fixed-point for open ball and bijective uniformly continuous function?
No, for instance a piecewise-linear (say with two pieces) bijection $f:(0,1)\to(0,1)$ whose graph lies on one side of the diagonal $y=x$.


