Is there a measure which allows me to tell how closely something is to an ellipse?

Question

Roundness is the measure of how closely the shape of an object approaches that of a circle.

I am trying to find a similar measure which shows how closely is something to an ellipse. Is there any similar measure?

Answer 1

Here's an idea based on the definition of roundness.

Let $C$ be a simple closed curve in $\mathbb{A}^2$, let $E_{in}$ be the set of ellipses contained in the interior of $C$, and let $E_{out}$ be the set of ellipses contained in the exterior of $C$. Then you can define your 'ellipseness' to be $$ \sup\left\{\frac{\operatorname{area} (A)}{\operatorname{area}(B)}: A\in E_{in}, B\in E_{out}\right\}. $$