what just happened?

content/Math401/Math401_P1_3.md

@@ -205,6 +205,22 @@ The goal is the same as the Gromov's version, first we bound the probability of
 
 Let $f:S^n\to \mathbb{R}$ be a 1-Lipschitz function.
 
+Consider the two sets of points on the sphere $S^n$ with radius $\sqrt{n}$:
+
+$$
+\Omega_+=\{x\in S^n: f(x)\leq a_0-\epsilon\}, \Omega_-=\{x\in S^n: f(x)\geq a_0+\epsilon\}
+$$
+
+Note that $\Omega_+\cup \Omega_-$ is the whole sphere $S^n(\sqrt{n})$.
+
+By the Levy's isoperimetric inequality, we have
+
+$$
+\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\epsilon))\leq \operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\Omega_+))+\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\Omega_-))
+$$
+
+
+
 We define $\kappa_n(\epsilon)$ as the following:
 
 $$