Timeline for Is the derivative the unique operation on points in the plane that preserves convexity?
Current License: CC BY-SA 4.0
11 events
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| Aug 30, 2021 at 15:15 | comment | added | Chris H | I'd prefer to leave the question up still I think, since it really is the general behaviour for all $n$ that I'm interested in. However, I would accept any answer which can characterise the derivative using these kind of geometric considerations (for instance partial unitary invariance, as submultisets of $S^2$). | |
| Aug 29, 2021 at 22:11 | comment | added | Gabriel C. Drummond-Cole | without something like the multiplicity condition in the question, "surjectivity + invariance" as suggested by fedja looks like it's probably possible with a weighted average of derivative with center of mass (I didn't check surjectivity rigorously). | |
| Aug 29, 2021 at 21:16 | comment | added | Gabriel C. Drummond-Cole | I will formulate this as an answer if @ChrisH thinks it is sufficient, but I tend to agree with fedja here. | |
| Aug 29, 2021 at 20:37 | comment | added | fedja | @მამუკაჯიბლაძე Formally yes, but really it is just another example showing that more conditions are required. I would suggest to combine surjectivity and invariance under complex linear mappings. Then for $n=2$, the midpoint becomes the only option and that is the derivative. But I'm pessimistic about large $n$ even after that. The actual question here seems to be "Is there any list of natural (whatever that means) properties to require to get an affirmative answer?" | |
| Aug 29, 2021 at 17:28 | comment | added | მამუკა ჯიბლაძე | @GabrielC.Drummond-Cole This is an answer, is not it? | |
| Aug 29, 2021 at 16:13 | history | edited | Gabriel C. Drummond-Cole | CC BY-SA 4.0 |
fixed edge case
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| Aug 29, 2021 at 4:42 | comment | added | Gabriel C. Drummond-Cole | Let $f$ be an odd function from $\mathbb{R}$ to $(-1,1)$ (e.g. $2\arctan(x)/\pi$). Then for $n=2$ take $\{x,y\}\mapsto \{((1+f(\mathrm{Re}(x) - \mathrm{Re}(y)))x + (1-f(\mathrm{Re}(x) - \mathrm{Re}(y)))y)/2\}$. For $f=0$ you get the derivative. | |
| Aug 29, 2021 at 1:50 | comment | added | Chris H | Ahh thanks, I added another condition, really I want to know if the derivative is uniquely defined by its metric behaviour on zeros (multisets), but I'm not sure how to best phrase that precisely. | |
| Aug 29, 2021 at 1:47 | history | edited | Chris H | CC BY-SA 4.0 |
Adding nontriviality conditions
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| Aug 29, 2021 at 1:38 | comment | added | fedja | Well, you, probably, want to add some extra assumptions. Otherwise something as boring as, say, center of mass repeated $n-1$ times works... | |
| Aug 28, 2021 at 23:33 | history | asked | Chris H | CC BY-SA 4.0 |