Quantitative Isoperimetric Inequalities on the Real Line
Annales mathématiques Blaise Pascal, Volume 18 (2011) no. 2, pp. 251-271.

In a recent paper A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli have shown that, in the Gauss space, a set of given measure and almost minimal Gauss boundary measure is necessarily close to be a half-space.

Using only geometric tools, we extend their result to all symmetric log-concave measures on the real line. We give sharp quantitative isoperimetric inequalities and prove that among sets of given measure and given asymmetry (distance to half line, i.e. distance to sets of minimal perimeter), the intervals or complements of intervals have minimal perimeter.

Dans un récent papier, A. Cianchi, N. Fusco, F. Maggi, et A. Pratelli ont montré que, dans l’espace de Gauss, un ensemble de mesure donnée et de frontière de Gauss presque minimal est nécessairement proche d’être un demi-espace.

En utilisant uniquement des outils géométriques, nous étendons leur résultat au cas des mesures log-concaves symétriques sur la droite réelle. On donne des inegalités isopérimétriques quantitatives optimales et l’on prouve que parmi les ensembles de mesure donnée et d’asyḿétrie donnée (distance à la demi-droite, i.e. distance aux ensembles de périmètre minimal), les intervalles ou les complémentaires d’intervalles ont le plus petit périmètre.

DOI: 10.5802/ambp.299
Classification: 26B15,  49Q15
Keywords: Isoperimetric inequalities, Asymmetry, Log-concave measures, Gaussian measure.
Yohann de Castro 1

1 Institut de Mathématiques de Toulouse (UMR CNRS 5219) Université de Toulouse III - Paul Sabatier 118, route de Narbonne, 31062 Toulouse, France.
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Yohann de Castro. Quantitative Isoperimetric Inequalities on the Real Line. Annales mathématiques Blaise Pascal, Volume 18 (2011) no. 2, pp. 251-271. doi : 10.5802/ambp.299. https://ambp.centre-mersenne.org/articles/10.5802/ambp.299/

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