# Löwner-John ellipsoids

So far, everything was composed of definitions and trivial generalities. Now comes something deceptively simple but more substantial.

This is how Gromov introduces the maximal volume ellipsoid in his paper “Hilbert volume in metric spaces I”. The result is easy to state: for every origin-symmetric convex body $K\subset \mathbb R^n$ there exists a unique ellipsoid $E$ of maximal volume among the ellipsoids contained in $K$, and it satisfies $E\subset K\subset \sqrt{n}E$. The constant $\sqrt{n}$ is sharp: it is attained, for example, when $K$ is a cube.

The story of the maximum-volume ellipsoid is presented by Martin Henk in “Löwner-John ellipsoids”. In my experience, the name of Charles Loewner is not attached to this concept as often as Fritz John’s. From my undergraduate years I associated Loewner only with the parametric method of representing conformal maps, which he used to prove $|c_3|\le 3$, the first hard case of Bieberbach’s conjecture $|c_n|\le n$.

Henk mentions Loewner’s stay at Syracuse University, but does not specify the term, which was 1946–1951. A longtime chair of Syracuse math department Donald Kibbey recalled in 1980:

Loewner went to Stanford, Bers went to NYU and then Columbia, Gelbart went to Yeshiva, Cairns to Illinois, Milgram to Minnesota, Rosenbloom to Minnesota and then Columbia, Lorentz to Texas, Chung to Stanford, Greub to Toronto, Gilchrist to IBM, Rheinboldt to Maryland, Mostow to Hopkins and then Yale, Rosskopf to Columbia, Robert Davis to Illinois, Donald Austin and Mark Mahowald to Northwestern, Kleinfeld to Iowa, Ryser to Cal Tech, and Selberg to the Institute for Advanced Study, and Erdős to the world…. It’s too bad we didn’t keep some of them, I of course had my druthers, some people in there—some were magnificent. But think how much we helped the other places, Stanford, in particular, two of the finest.