When I learn the theorem about Camanato Criterion, I need to verify the following fact: For unit sphere $B$ in $\mathbb{R}^n$, there exists a constant $C(n)$ such that $$Lebesgue~measure~of(B\cap B(x,r))\geq C(n)r^n$$ for any $x\in B$ and $0<r<2$.
I intuitively feel that this conclusion is correct, and think it has something to do with the smoothness of the boundary of the unit sphere, but I don't know how to prove it strictly.
Furthermore, I also guess that there must exists an optimal $C(n)$, but how can we find it.
I do hope somebody can help me! Thanks!
Let $\vert A \vert$ denote the Lebesgue measure of a (measurable) set $A\subset \mathbb R^n$ and denote by $\omega_n$ the Lebesgue measure of $B_1$ i.e. $\omega_n:= \vert B_1\vert$.
Suppose that $x\in B_1$, $\rho \in (0,2)$, and let $x_\rho := (1-\rho/2)x$. Observe that $B_{\rho/2}(x_\rho)\subset B_1 \cap B_\rho(x)$. Indeed, $$\vert x_\rho \vert = \bigg (1 - \frac \rho 2 \bigg ) \vert x \vert <1, $$ so $x_\rho \in B_1$ and $$ \vert x_\rho-x\vert =\frac \rho 2 \vert x\vert <\rho, $$ so $x_\rho \in B_\rho(x)$. Hence, $$ \vert B_1 \cap B_\rho(x) \vert \geqslant \vert B_{\rho/2}(x_\rho) \vert = \omega_n \bigg ( \frac \rho 2 \bigg )^n= 2^{-n} \omega_n \rho^n. $$ This proves the inequality with $C(n) = 2^{-n} \omega_n$.
Moreover, I claim that this constant is in fact optimal. Indeed, when $\rho\geqslant 1+\vert x \vert$ then $B_1\cap B_\rho(x) = B_1$, so $$ \frac{\vert B_1\cap B_\rho(x) \vert}{\rho^n} = \frac{\omega_n}{\rho^n} \to 2^{-n}\omega_n $$ as $\rho \to 2^-$.