Given a convex polygon in the plane, consider the smallest-area ellipse that contains this polygon. This is the "minimal volume covering ellipsoid" or "minimal volume enclosing ellipsoid" (MVEE), and apparently it's been studied quite a bit. Is there a bound on the volume of this MVEE? In other words, what is the following?
$$\mbox{sup}\{\mbox{Vol}(E):E\mbox{ is an MVEE for some convex }K\subset\mathbb{R}^{2}\mbox{ satisfying Vol}(K)=1\}$$
More generally, of course, I wonder what this value would be in any dimension. I'm interested in convex polygons and convex bodies. For general polygons the answer is no, as shown below.
Sounds like you want to look into the theory of Löwner-John ellipsoids (which states that the relation between the (by volume) largest possible inscribed ellipsoid inside a convex set $K$, and an outer circumscribing ellispoid, is bounded by $n$ ($\sqrt{n}$ for symmetric bodies). From there, you would get a volume bound.)