2021
DOI: 10.5186/aasfm.2021.4666
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Quantitative lower bounds to the Euclidean and the Gaussian Cheeger constants

Abstract: We provide a quantitative lower bound to the Cheeger constant of a set Ω in both the Euclidean and the Gaussian settings in terms of suitable asymmetry indexes. We provide examples which show that these quantitative estimates are sharp.

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Cited by 5 publications
(4 citation statements)
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“…Therefore, all of our results apply in this setting. While the existence of 1‐Cheeger sets was already known, see [51, 86], and the clustering isoperimetric problem has been recently addressed in [106], the Cheeger cluster problem for N>1$N>1$ had never been treated. The relation with the prescribed curvature functional had been studied in [51], while, up to our knowledge, the relation with the first eigenvalue of the Dirichlet p$p$‐Laplacian had never been proved, but only quickly observed in [86] for p=2$p=2$ (the Ornstein–Uhlenbeck operator).…”
Section: Applicationsmentioning
confidence: 99%
See 1 more Smart Citation
“…Therefore, all of our results apply in this setting. While the existence of 1‐Cheeger sets was already known, see [51, 86], and the clustering isoperimetric problem has been recently addressed in [106], the Cheeger cluster problem for N>1$N>1$ had never been treated. The relation with the prescribed curvature functional had been studied in [51], while, up to our knowledge, the relation with the first eigenvalue of the Dirichlet p$p$‐Laplacian had never been proved, but only quickly observed in [86] for p=2$p=2$ (the Ornstein–Uhlenbeck operator).…”
Section: Applicationsmentioning
confidence: 99%
“…Because of its numerous applications, several authors have been drawn to the subject and started to investigate the constants and the above‐mentioned links with other problems in several frameworks: weighted Euclidean spaces [7, 26, 96, 114]; finite‐dimensional Gaussian spaces [51, 86]; anisotropic Euclidean and Riemannian spaces [8, 21, 49, 89]; the fractional perimeter [28] or nonsingular nonlocal perimeter functionals [100]; Carnot groups [108]; RCD$\mathsf {RCD}$‐spaces [70, 71]; and lately smooth metric‐measure spaces [1].…”
Section: Introductionmentioning
confidence: 99%
“…Therefore, all of our results apply in this setting. While the existence of 1-Cheeger sets was already known, see [50,81], and the clustering isoperimetric problem has been recently addressed in [101], the cluster problem for N > 1 had never been treated. The relation with the prescribed curvature functional had been studied in [50], while, up to our knowledge, the relation with the first eigenvalue of the Dirichlet p-Laplacian had never been proved, but only quickly observed in [81] for p = 2 (the Ornstein-Uhlenbeck operator).…”
Section: Euclidean Spaces With Densitymentioning
confidence: 99%
“…Since then, the Euclidean case has been the most studied, and we refer to the two surveys [87,108], but many different contexts have been object of research, such as: weighted Euclidean spaces [7,26,91,111]; finite dimensional Gaussian space [50,81]; anisotropic Euclidean and Riemannian spaces [8,22,48,83]; the fractional perimeter [28] or non-singular non-local perimeter functionals [97]; Carnot groups [104]; RCD-spaces [67,68]; and lately smooth metric-measure spaces [1].…”
Section: Introductionmentioning
confidence: 99%