Given $\epsilon > 0$ and $f : [0, 1]^{\omega} \rightarrow [0, 1]^{\omega}$, can we find $x$ such that $x \in \textrm{Conv}\left( \left\{f(y) : ||y - x||\_{\infty} < \epsilon\right\}\right)$? 


In finite dimensions this is straightforwardly equivalent to Brouwer's fixed point theorem. In the infinite dimensional case they are not equivalent, and I don't even know whether to expect the claim to be true or false (I was expecting it to be much more straightforward to settle). 


I would actually be happy to solve the problem for the simpler case of maps $f : [0, 1]^{\omega} \rightarrow \left\{0, 1\right\}^{\omega}$, since I'm imagining each coordinate of $f$ as returning the truth of a certain predicate applied to $x$. And we could just as well work with the discrete setting, of $f : \left\{0, 1, \ldots, m\right\}^{\omega} \rightarrow \left\{0, m\right\}^{\omega}$. 


I'm aware of [this result](http://arxiv.org/pdf/math/0208141.pdf), but it doesn't seem to help very much here.


Edit: I would actually be happy to find an $x$ which isn't separated from $\left\{ f(y) : ||y - x||\_{\infty} < \epsilon\right\}$ by any hyperplane parallel to the coordinate axes. This seems like it should be much easier.



An approximate infinite-dimensional fixed point theorem
In fact the property fails on any non-compact closed convex set $K$ of a Banach space : there exists a Lipschitz map $f:K\to K$ with no approximate fixed point, that is $\inf\_{x\in K}\|f(x)-x\|=\delta > 0$ (See [*Geometric Nonlinear Functional Analysis- Part 1*](http://books.google.it/books?id=lXZ95EKwjYUC&printsec=frontcover&dq=Geometric+Nonlinear+Functional+Analysis,+Part++1+++Benyamini,+Lindenstrauss&hl=it&sa=X&ei=MgluUpnvA6aN0AXFyICwCQ&ved=0CDQQ6AEwAA#v=onepage&q&f=false) by Benyamini and Lindenstrauss, Thm 3.4 ). If we take $\epsilon:=\delta/\mathrm{lip}(f)$ there holds $\operatorname{co}f\big(B(x,\epsilon)\big)\subset B\big(f(x), \delta\big)\not\ni x ,$ for any $x\in K$.


