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 $k$ 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?

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?

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

Such that the convex hull of $D(S)$ is contained in the convex hull of $S$.

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?

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 $k$ 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?