Löwner–John theorem

Remarks

One may take $A$ symmetric positive definite, by the polar decomposition. Existence follows from compactness: the pairs $(c, A)$ with $A$ positive semidefinite and $c + AB \subseteq K$ form a compact set, and volume is continuous. For uniqueness, let $c_1 + A_1 B$ and $c_2 + A_2 B$, with $A_i$ positive definite, both have maximal volume. Then $\frac{c_1 + c_2}{2} + \frac{A_1 + A_2}{2} B$ also lies in $K$. Its volume is strictly larger unless $A_1 = A_2$, by the strict log-concavity of the determinant on positive definite matrices. If $A_1 = A_2$ but $c_1 \neq c_2$, the convex hull of the two ellipsoids contains a larger ellipsoid, stretched in the direction $c_2 - c_1$. John characterized the case $E = B$ by contact points. For $K \supseteq B$, the ball $B$ is the maximal ellipsoid if and only if there are $u_1, \dots, u_m \in \partial K$ with $\abs{u_i} = 1$ and weights $c_i > 0$ such that $\sum_i c_i u_i = 0$ and $\sum_i c_i u_i u_i^T = I_n$. The bounds in 2 and 3 follow from this. The factor $n$ in 2 is attained by a regular simplex. The factor $\sqrt{n}$ in 3 is attained by the cube $[-1, 1]^n$, whose maximal ellipsoid is $B$ and which contains the point $(1, \dots, 1)$ of norm $\sqrt{n}$.

Going deeper

For a student

Part 3 says that every $n$-dimensional normed space is at Banach–Mazur distance at most $\sqrt{n}$ from Euclidean space, a basic fact of the local theory of Banach spaces. The contact point condition in John's characterization is a decomposition of the identity. Ball combined it with the Brascamp–Lieb inequality to prove his reverse isoperimetric inequality. In optimization, rounding algorithms for convex bodies compute approximate John ellipsoids.