Assume a Riemannian \(d\)-manifold \((M,g)\) with non-negative sectional curvature . Let \(B_r\subseteq M\) be a geodesic ball with radius \(r>0\) and \(S\subseteq B_r\) a subset with positive measure. Then for every point \(p\in B_r\) there is geodesic \(\gamma\colon I\to M\) starting in \(p\) and lying in \(B_r\), such that
\begin{equation*} \Vol(S) \le C r^{d-1} \lvert \gamma(I)\cap S\rvert \end{equation*}where \(C>0\) is constant depending only on the dimension \(d\).
Consider a normal coordinates centered at \(p\). Then in this coordinates we have
\begin{equation*} \lvert B_r\cap S\rvert = \int_{\mathbb{R}^d} 𝟙_{B_r\cap S}(x) \sqrt{\det g_{ij}} \,dx. \end{equation*}Since the sectional curvature is non-negative, we estimate the above integral applying … and obtain
\begin{equation*} \lvert B_r\cap T\rvert \le \int_{\mathbb{R}^d} 𝟙_{B_r\cap T}(x) \,dx. \end{equation*}Using the same arguments as in (0x6698db36) , we find a direction \(\eta \in \mathbb{S}^{d-1}\), such that
\begin{equation*} \lvert B_r\cap S\rvert \le \sigma_{d-1} (2r)^{d-1}\int_{0}^{\infty} 𝟙_{B_r\cap T}(x+\rho\eta) \,d\rho, \end{equation*}where \(\sigma_{d-1}=\lvert \mathbb{S}^{d-1}\rvert\).
The identity \(\lvert \gamma(I)\cap S\rvert = \int_{0}^{\infty} 𝟙_{B_r\cap T}(x+\rho\eta) \,d\rho\) completes the proof.
- We are able to move the condition on the sectional curvature by considering small enough radius. The exact value will then depend on the sectional curvature bounds.