There is a very nice short proof of John's ellipsoid theorem by Gruber and Schuster: https://www.dmg.tuwien.ac.at/gruber/gruber_arbeiten/johnellipsoid.pdf http

There is a very nice short proof of John's ellipsoid theorem by Gruber and Schuster: https://www.dmg.tuwien.ac.at/gruber/gruber_arbeiten/johnellipsoid.pdf https://www.dmg.tuwien.ac.at/gruber/gruber_arbeiten/johnelli... — one elegant trick I remember from there was that the value of a quadratic form with matrix A on vectors u and v (^T for transpose): u^T A v is interpreted as the dot product between the matrix A and the tensor product u v^T, A • (u v^T) — and dot product • on matrices is just from them being n×n vectors. With that a lot of things are really nice now, e.g. interiors of ellipsoids correspond to intersections of halfspaces of matrices with the positive semidefinite cone. And halfspaces are simple to reason about and intersect! This trick is also implicitly in the parent post, of course.