The Paulsen problem made simple

The Paulsen problem was once thought to be one of the most intractable problems in frame theory. It was resolved in a recent tour-de-force work of Kwok, Lau, Lee and Ramachandran. In particular, they showed that every ϵ -nearly equal norm Parseval frame in d dimensions is within squared distance O ( ϵd 13/2 ) of an equal norm Parseval frame. We give a dramatically simpler proof based on the notion of radial isotropic position, and along the way show an improved bound of O ( ϵd 2 ).