# Brouwer fixed-point for open ball and bijective uniformly continuous function?

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 $B^-$?

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 $y=x$.
• So now I'll start thinking again about which natural extra conditions on $f$ would ensure a fixed point in the open ball... – Frank Waaldijk May 6 '18 at 16:08