Timeline for Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
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| Mar 20, 2015 at 19:30 | comment | added | Igor Rivin | Ah, did not notice the 1-1 hypothesis... | |
| Mar 20, 2015 at 18:25 | comment | added | Ramiro de la Vega | Indeed, the interesting case is when the function is not one to one (space-filling curves). For a one-to-one function from $\mathbb{R}$ to $\mathbb{R}^2$, the image of each $[-n, n]$ must be a convex set homeomorphic to $[0,1]$, hence a line segment. | |
| Mar 20, 2015 at 18:12 | comment | added | Joonas Ilmavirta | The theorem reads "Let $V$ and $W$ be real vector spaces. Any one-to-one mapping $f:V\to W$ which preserves convexity is collinearity-preserving." The OP did not assume the function to be one-to-one, so this does not fully answer the question. | |
| Mar 20, 2015 at 18:03 | history | answered | Igor Rivin | CC BY-SA 3.0 |