# fa.functional analysis – Is the derivative the unique operation on points in the plane that preserves convexity?

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 $$ngeq 2$$.

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