John's ellipsoid theorem

Sat, 25 Apr 2026 00:00:00 +0000

John’s ellipsoid theorem is a clean high-dimensional convex geometry fact that shows up in a lot of different places. Informally, it says:

Every $n$-dimensional symmetric convex body is an ellipsoid, up to a $\sqrt n$ scaling.

This is quite nice and convenient because ellipsoids are basically “$\ell_2$” objects, which makes them a lot easier to do math with than arbitrary convex bodies. There is a long list of applications of John’s theorem that I won’t cover here, but it recently came up for me in three different contexts.

Theory Bubbles on Sidhanth Mohanty

John's ellipsoid theorem