John ellipsoid

Summary

The John ellipsoid of a convex body $K$ in $\R^n$ with nonempty interior is the ellipsoid of largest volume contained in $K$. It exists and is unique by the Löwner–John theorem, and $K$ is contained in its dilate by the factor $n$ about its center. It gives a canonical affine position of $K$. It is the starting point of many estimates in convex geometry and the geometry of normed spaces. Its dual partner is the Löwner ellipsoid, the ellipsoid of least volume containing $K$.