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Let $C(n)$ be the space of multisets of size $n$ of points in the Euclidean plane, topologised appropriately, and consider a surjective continuous map: $$D:C(n)\rightarrow C(n-1)$$

Such that the convex hull of $D(S)$ is contained in the convex hull of $S$, and if a multiset $S$ has $x$ occurring $k>0$ times, then $D(S)$ has $x$ occurring $k-1$ times.

If we identify the plane with $\mathbb{C}$, multisets with polynomials, then by the Gauss-Lucas theorem, the derivative is a map with this property, for $n\geq 2$.

Does this property characterise the derivative? So for $n\geq 2$, is this the only continuous map between these spaces with this property?

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    $\begingroup$ Well, you, probably, want to add some extra assumptions. Otherwise something as boring as, say, center of mass repeated $n-1$ times works... $\endgroup$
    – fedja
    Commented Aug 29, 2021 at 1:38
  • $\begingroup$ 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. $\endgroup$
    – Chris H
    Commented Aug 29, 2021 at 1:50
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    $\begingroup$ 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. $\endgroup$
    – Gabriel C. Drummond-Cole
    Commented Aug 29, 2021 at 4:42
  • $\begingroup$ @GabrielC.Drummond-Cole This is an answer, is not it? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Aug 29, 2021 at 17:28
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    $\begingroup$ @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?" $\endgroup$
    – fedja
    Commented Aug 29, 2021 at 20:37

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