Uniqueness of Lp Minkowski problem in the supercritical range

The uniqueness of the $L_p$-Minkowski problem has been a long standing problem in convex geometry. In the groundbreaking paper by Brendle-Choi-Daskalopoulos (Acta Math, {\bf219}, 2017), a full uniqueness result was shown for the subcritical exponents $p\in (-n-1,1]$. In the supercritical range, the uniqueness problem is much more complicated, even on the planar case $n=1$. One of the famous results was shown by Andrews in (J. Amer. Math. Soc., {\bf16}, 2003), where he established that the uniqueness holds in the range $p\in (-7,-2)$ and fails to hold for the other supercritical exponents $p\in (-\mathrm{\infty },-7)$. In this paper, we study the same uniqueness problem in the full supercritical range $p\in (-2n-5,-n-1)$ for all higher dimensional cases $n\ge 2$. We will prove that for $p\in (-2n-5,-n-1)$, the unique strongly symmetric solution is given by the unit sphere ${\mathrm{\S }}^n$. The uniqueness range $(-2n-5,-n-1)$ is optimal due to our recent preprint (arXiv: 2104.07426), where non-spherical strongly symmetric solutions have been constructed for all $p\in (-\mathrm{\infty },-2n-5)$. When considering general solutions which may not be symmetric, the uniqueness set $\mathrm{\Gamma }$ of $p$ for which the uniqueness holds, is shown to be both relatively open and closed in the full interval $(-2n-5,-n-1)$.