Update Math401_P1_3.md

content/Math401/Math401_P1_3.md

@@ -108,4 +108,20 @@ $$
 \kappa_n(\epsilon)=\frac{\int_\epsilon^{\frac{\pi}{2}}\cos^{n-1}(t)dt}{\int_0^{\frac{\pi}{2}}\cos^{n-1}(t)dt}
 $$
 
+$a_0$ is the **Levy mean** of function $f$, that is the level set of $f^{-1}:\mathbb{R}\to S^n$ divides the sphere into equal halves, characterized by the following equality:
+
+$$
+\mu(f^{-1}(-\infty,a_0])\geq \frac{1}{2} \text{ and } \mu(f^{-1}[a_0,\infty))\geq \frac{1}{2}
+$$
+
 Hardcore computing may generates the bound but M. Gromov did not make the detailed explanation here.
+
+#### Detail proof by Takashi Shioya
+
+
+
+## References
+
+- [High-dimensional probability by Roman Vershynin](https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-2.pdf)
+- [Metric Structures for Riemannian and Non-Riemannian Spaces by M. Gromov](https://www.amazon.com/Structures-Riemannian-Non-Riemannian-Progress-Mathematics/dp/0817638989/ref=tmm_hrd_swatch_0?_encoding=UTF8&dib_tag=se&dib=eyJ2IjoiMSJ9.Tp8dXvGbTj_D53OXtGj_qOdqgCgbP8GKwz4XaA1xA5PGjHj071QN20LucGBJIEps.9xhBE0WNB0cpMfODY5Qbc3gzuqHnRmq6WZI_NnIJTvc&qid=1750973893&sr=8-1)
+- [Metric Measure Geometry by Takashi Shioya](https://arxiv.org/pdf/1410.0428)