Uniqueness of $S_2$-isotropic solutions to the isotropic $L_p$ Minkowski problem
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914031408775168 |
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| author | Wan, Yao |
| author_facet | Wan, Yao |
| contents | 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 $λ_2(-L_K)\geq \frac{n-1}{2n-1+p}$. Furthermore, we extend this uniqueness result to the range $-2n-1 \leq p<-n$ with $λ_2(-L_K)\geq \frac{-p-1}{n-1}$, assuming the origin-centred condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08588 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniqueness of $S_2$-isotropic solutions to the isotropic $L_p$ Minkowski problem Wan, Yao Differential Geometry Analysis of PDEs Metric Geometry 53A07, 35A02, 52A20 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 $λ_2(-L_K)\geq \frac{n-1}{2n-1+p}$. Furthermore, we extend this uniqueness result to the range $-2n-1 \leq p<-n$ with $λ_2(-L_K)\geq \frac{-p-1}{n-1}$, assuming the origin-centred condition. |
| title | Uniqueness of $S_2$-isotropic solutions to the isotropic $L_p$ Minkowski problem |
| topic | Differential Geometry Analysis of PDEs Metric Geometry 53A07, 35A02, 52A20 |
| url | https://arxiv.org/abs/2509.08588 |