Cheeger's isoperimetric problem for Gaussian mixtures
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912907507269632 |
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| author | Liehr, Lukas |
| author_facet | Liehr, Lukas |
| contents | In any dimension $n$, we determine the Cheeger constant and the Cheeger sets of the Gaussian mixture $μ(x) = pγ(x-a) + (1-p)γ(x-b)$, where $p \in [0,1]$, $a,b \in \mathbb{R}^n$, and $γ: \mathbb{R}^n \to (0,\infty)$ denotes a Gaussian. In particular, we characterize the Cheeger sets for $μ$ in terms of specific half-spaces perpendicular to $a-b$, thereby confirming the conjectured solution to the Cheeger problem for Gaussian mixtures. Finally, we study the regime of parameters $p,a,b$ in which $μ$ admits a unique Cheeger set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14724 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cheeger's isoperimetric problem for Gaussian mixtures Liehr, Lukas Functional Analysis Classical Analysis and ODEs Probability 49J40, 49Q20, 60E15 In any dimension $n$, we determine the Cheeger constant and the Cheeger sets of the Gaussian mixture $μ(x) = pγ(x-a) + (1-p)γ(x-b)$, where $p \in [0,1]$, $a,b \in \mathbb{R}^n$, and $γ: \mathbb{R}^n \to (0,\infty)$ denotes a Gaussian. In particular, we characterize the Cheeger sets for $μ$ in terms of specific half-spaces perpendicular to $a-b$, thereby confirming the conjectured solution to the Cheeger problem for Gaussian mixtures. Finally, we study the regime of parameters $p,a,b$ in which $μ$ admits a unique Cheeger set. |
| title | Cheeger's isoperimetric problem for Gaussian mixtures |
| topic | Functional Analysis Classical Analysis and ODEs Probability 49J40, 49Q20, 60E15 |
| url | https://arxiv.org/abs/2602.14724 |