We prove that the sharp Buser's inequality obtained in the framework of $\mathsf{RCD}(1,\infty )$ spaces by the first two authors [29] 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.

中文翻译：

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Cheeger和Buser不等式中的等式案例 $\mathsf{刚果民盟}$ 空格

我们证明了在 $\mathsf{刚果民盟}（\mathrm{1个}，\infty ）$前两个作者[29]提出的空间是刚性的，即当且仅当空间同构地分裂为高斯时，才能获得相等性。即使在平滑设置下，结果也是新的。我们还表明，在以下情况下，切格不等式的平等是永远无法实现的$\mathsf{刚果民盟}（ķ，\infty ）$ 具有有限直径或正曲率的空间，我们提供Ricci曲率限定在下面的空间的几个示例，在这些条件下不满足这些假设并且获得了相等性。