On the uniqueness of even $L^p$ Minkowski problem

We prove that there is a unique $p_0\in [0,1)$, which can be characterized by the eigenvalue of Hilbert operator related to a convex body, that the even $L^p$ Minkowski problem has a unique solution for $p\geq p_0$, and the uniqueness fails for infinitely many convex bodies if $p<p_0$. The previous results by many experts in the field assert that the uniqueness holds for $p>p_0$.