Let $f$ be an essential map of $S^{n-1}$ into itself (i.e., $f$ is not homotopic to a constant mapping) admitting an extension
mapping the closed unit ball $\overline B^n$ into $\mathbb{R}^n$ .
Then, for every interior point $y$ of $B^n$ , there exists a point
$x$ in $f^{-1}(y)$ such that the image of no neighborhood of $x$ is contained in a coordinate half space with $y$ on its boundary.
Under additional conditions, the image of a neighborhood of $x$ covers a neighborhood of $y$ . Differential versions are valid
for quasianalytic functions. These results are motivated by
game-theoretic considerations.