John ellipsoid

Remarks

The Löwner–John theorem shows that this ellipsoid exists and is unique. The ellipsoid of minimal volume containing $K$ is the Löwner ellipsoid (or Löwner–John ellipsoid) of $K$.

Example: $K = [-1, 1]^2$. Its John ellipsoid is the unit disk $B$. The problem is invariant under the symmetries of the square, so by uniqueness the maximal ellipse is invariant under them too. An ellipse invariant under rotation by $90°$ about the origin is a disk centered at $0$, and the largest such disk in $K$ is $B$.

Non-example: in the rectangle $K = [-2, 2] \times [-1, 1]$ the unit disk lies in $K$ but is not the John ellipsoid. The ellipse $\set{(x, y) : x^2/4 + y^2 \le 1}$ also lies in $K$ and has area $2\pi > \pi$.

The condition that $K$ has nonempty interior is needed: otherwise every ellipsoid in $K$ would be degenerate, and no ellipsoid in the above sense lies in $K$.