John's ellipsoid theorem
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. ...