We study decreasing rearrangements of functions defined on (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by $K>0$ and dimension bounded above by $N\in (1,\infty )$ in a synthetic sense, the so called $\mathrm{CD}(K,N)$ spaces. We first establish a Polya-Szego type inequality stating that the $W^{1,p}$-Sobolev norm decreases under such a rearrangement and apply the result to show sharp spectral gap for the p-Laplace operator with Dirichlet boundary conditions (on open subsets), for every $p\in (1,\infty )$. This extends to the non-smooth setting a classical result of Bérard-Meyer [14] and Matei [41]; remarkable examples of spaces fitting our framework and for which the results seem new include: measured-Gromov Hausdorff limits of Riemannian manifolds with Ricci $\ge K>0$, finite dimensional Alexandrov spaces with curvature$\ge K>0$, Finsler manifolds with Ricci $\ge K>0$.

In the second part of the paper we prove new rigidity and almost rigidity results attached to the aforementioned inequalities, in the framework of $\mathrm{RCD}(K,N)$ spaces, which are interesting even for smooth Riemannian manifolds with Ricci $\ge K>0$.

我们研究在Ricci曲率下限为（）的（可能是非平滑）度量度量空间上定义的函数的递减重排 $ķ>0$ 和尺寸受上述限制 $ñ\in （\mathrm{1个}，\infty ）$ 在综合意义上，所谓的 $\mathrm{光盘}（ķ，ñ）$空格。我们首先建立Polya-Szego型不等式，说明${\mathrm{w\; ^}}^{\mathrm{1个}，p}$-Sobolev范数在这种重排下减小，并将结果应用到显示Dirichlet边界条件（在开放子集上）的p -Laplace算子的尖锐谱隙$p\in （\mathrm{1个}，\infty ）$。这延伸到非平滑设置的Bérard-Meyer[14]和Matei [41]的经典结果。符合我们框架的空间的显着示例，其结果似乎是新的，包括：黎曼流形与Ricci的实测格罗莫夫Hausdorff极限$\ge ķ>0$曲率的有限维亚历山大空间$\ge ķ>0$，Ricci的Finsler流形 $\ge ķ>0$。

在本文的第二部分中，我们证明了新的刚性和几乎刚性的结果附加到上述不等式上， $\mathrm{刚果民盟}（ķ，ñ）$ 空间，即使对于带有Ricci的光滑黎曼流形也很有趣 $\ge ķ>0$。