query: Brouwer fixed-point theorem on non-convex sets
positive_passages.docid: 23875
positive_passages.text: Why is convexity a requirement for Brouwer fixed points? Shouldn't "no holes" be good enough?
negative_passages.docid: 71619
negative_passages.text: Proving that a convex function is Lipschitz
negative_passages.docid: 61122
negative_passages.text: A continuous mapping is determined by its values on a dense set
negative_passages.docid: 72659
negative_passages.text: How can we define convexity in one dimension?
negative_passages.docid: 146062
negative_passages.text: A "Cantor-Schroder-Bernstein" theorem for partially-ordered-sets
negative_passages.docid: 20614
negative_passages.text: Example of a function such that $\varphi\left(\frac{x+y}{2}\right)\leq \frac{\varphi(x)+\varphi(y)}{2}$ but $\varphi$ is not convex
negative_passages.docid: 64261
negative_passages.text: example of not surjective isometry on a bounded not closed set.
negative_passages.docid: 83784
negative_passages.text: Fixed Point of $x_{n+1}=i^{x_n}$
negative_passages.docid: 73074
negative_passages.text: $f \text{ convex }\Rightarrow \forall (x_0,f(x_0))\in \Gamma(f) \exists a : \forall x\in \mathbb{R}, f(x) \ge a(x-x_0)+f(x_0)$
negative_passages.docid: 128410
negative_passages.text: Showing that the set of fixed points is a smooth manifold
negative_passages.docid: 64197
negative_passages.text: Subgradients of non-convex functions
negative_passages.docid: 54222
negative_passages.text: Prove the map has a fixed point
negative_passages.docid: 65522
negative_passages.text: The wedge sum of two circles has fixed point property?
negative_passages.docid: 74352
negative_passages.text: A weak version of Markov-Kakutani fixed point theorem
negative_passages.docid: 42638
negative_passages.text: Showing that a function $f$ has a unique fixed point in a metric space.
negative_passages.docid: 40438
negative_passages.text: Proving that closed (and open) balls are convex
negative_passages.docid: 112362
negative_passages.text: fixed point problem for orthogonal transformation