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arXiv · 2008.12358

The equality case in Cheeger's and Buser's inequalities on $\mathsf{RCD}$ spaces

Nicolò De Ponti·Andrea Mondino·Daniele Semola
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Abstract

We prove that the sharp Buser's inequality obtained in the framework of $\mathsf{RCD}(1,\infty)$ spaces by the first two authors is rigid, i.e. equality is obtained if and only if the space splits isomorphically a Gaussian. The result is new even in the smooth setting. We also show that the equality in Cheeger's inequality is never attained in the setting of $\mathsf{RCD}(K,\infty)$ spaces with finite diameter or positive curvature, and we provide several examples of spaces with Ricci curvature bounded below where these assumptions are not satisfied and the equality is attained.

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Nicolò De Ponti, Andrea Mondino, Daniele Semola. 2020-08-27. The equality case in Cheeger's and Buser's inequalities on $\mathsf{RCD}$ spaces. https://doi.org/10.1016/j.jfa.2021.109022

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