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Results tagged with fixed-point-theorems
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user 31437
A fixed-point theorem is a result saying that a function $F$ will have at least one fixed point (a point $x$ for which $F(x) = x$), under some conditions on $F$ that can be stated in general terms.
1
vote
An approximate infinite-dimensional fixed point theorem
In fact there is a Lipschitz map which has no approximate fixed point in this sense.
Take some Lipschitz retraction $f$ from the interior of the cube into its exterior, and take $\epsilon$ much small …
- 777
4
votes
2
answers
424
views
An approximate infinite-dimensional fixed point theorem
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 finit …
- 777
3
votes
Accepted
Infinite-dimensional hex
It seems the claim is false, though I don't have a good sense for what the actual counterexample looks like. With luck, I've made an error somewhere.
Consider the normed space $\ell^{\infty}$. One c …
- 777
9
votes
1
answer
454
views
Infinite-dimensional hex
Suppose $n$ players take turns selecting vertices of the grid $[k]^n = \left\{0, 1, 2, \ldots, k-1\right\}^n$. Each player is assigned a pair of opposite faces of the grid, and wins the game if they c …
- 777