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

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 <a href="http://arxiv.org/pdf/math/0208141.pdf">this result</a>, but it doesn't seem to help very much here. Similar techniques can give you a point $x$ which is not separated from nearby $f(y)$ by any finitely supported hyperplane (or any hyperplane corresponding to an element of $\ell_{1}$), but the challenging separating hyperplanes are not of this form.