Return to Question

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, 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.

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, 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.

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 this result, 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.

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, 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.

If $\epsilon$ and $f : [0, 1]^k \rightarrow [0, 1]^k$ are arbitrary, it is easy to show that there is some $x \in [0, 1]^k$ such that $x$ is in the convex hull of points $f(y)$ with $||y - x||_{\infty} < \epsilon$.

Is this still true if $k$ is infinite? I.e., if $\epsilon$ and $f : [0, 1]^{\omega} \rightarrow [0, 1]^{\omega}$ are arbitrary, is there some $x \in [0, 1]^{\omega}$ such that $x$ is a convex combination of points $f(y)$ with $||y - x||_{\infty} < \epsilon$? (Note: I'm happy to take the closure of the convex hull under uniform convergence, which doesn't change the theorem, but not under pointwise convergence, which I believe would make the result easy.)

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 want to find such a fixed point because it implies the existence of a lookup table which approximately encodes all true facts about itself.

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, 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 most hyperplanes are not of this form.

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 this result, 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.