Given a triangle, there exists a unique circle passing through its $3$ vertices.
I wonder if more generally for any $n\geq3$ we can define a family $\mathcal{C}_n$ of convex closed curves in the plane with the following property:
For any convex (compact) polytope with $n$ vertices there exists a unique curve $C\in\mathcal{C}_n$ passing through the $n$ vertices.
In other words, I am looking for a generalization of the circumscribed circle to a triangle.
We can of course set $\mathcal{C}_3=\mathrm{Circles}$.
I thing it is well known that through $5$ points in generic position there is a unique conic. I guess that if the $5$ points are the vertices of a convex polygone then this conic is an ellipse. So I thing we can choose $\mathcal{C}_5=\mathrm{Ellipses.}$
I thing that we could set $\mathcal{C}_4=\mathrm{Ellipses}_D$ where $\mathrm{Ellipses}_D$ is the set of ellipses where the direction $D$ of the major axis is fixed.
More generally my guess is that we can define $\mathcal{C}_n$ has a subset of the the algebraic curve of even degree. I thing we need even degree in order to have a closed curve.
Any answer, complete or partial are welcome. I would like to have a better picture of what is reasonable to consider.