Uniqueness of $S_2$-isotropic solutions to the isotropic $L_p$ Minkowski problem

This paper investigates the spectral properties of the Hilbert-Brunn-Minkowski operator $L_K$ to derive stability estimates for geometric inequalities, including the local Brunn-Minkowski inequality. By analyzing the eigenvalues of $L_K$, we establish the uniqueness of $S_2$-isotropic solutions to the isotropic $L_p$ Minkowski problem in $\mathbb{R}^{n}$ for $\frac{1-3n^2}{2n}\leq p<-n$ with $\lambda_2(-L_K)\geq \frac{n-1}{2n-1+p}$. Furthermore, we extend this uniqueness result to the range $-2n-1 \leq p<-n$ with $\lambda_2(-L_K)\geq \frac{-p-1}{n-1}$, assuming the origin-centred condition.