Devesh Rajpal : Uniqueness of Lp-Minkowski problem near the affine exponent

We prove dimensional C0 estimates for solutions of the Lp-Minkowski problem in the entire supercritical range. This implies a spectral gap of the Brunn-Minkowski operator for the Lp Minkowski solutions. Furthermore, following methods from Ivaki-Milman and Du's results, this allows us to prove uniqueness of the Lp Minkowski problem in the supercritical range near the affine exponent p = -n-1 and importantly without any symmetry assumption.