I need a reference where to find the proof of the following theorem that I have found on wikipedia:
Let $M$ be an $n-$dimensional Riemannian closed compact manifold and let $p \in M$. Let $S(p)$ be the scalar curvature of $M$ at $p$. Then it holds $$ \frac{\operatorname{Vol}(B_\varepsilon(p)\subset M)}{\operatorname{Vol}(B_\varepsilon(0)\subset {\mathbb R}^n)}=1-\frac{S(p)}{6(n+2)}\varepsilon^2 + O(\varepsilon^4)$$ and $$\frac{\operatorname{Area} (\partial B_\varepsilon(p) \subset M)}{\operatorname{Area}(\partial B_\varepsilon(0)\subset {\mathbb R}^n)}= 1- \frac{S(p)} {6n} \varepsilon^2 + O(\varepsilon^4)$$ as $\epsilon \to 0^+$.
This would give me, if I am not misunderstanding anything, quantitative uniform bounds (both lower and upper) on the volume of a ball and of a sphere when $\epsilon$ is sufficiently small.
While there must be other sources, one can prove this by integrating (using polar coordinate in $n$-dimension) an expansion formula as in the book "Lectures on differential geometry" by R. Schoen and S-T. Yau, Chapter 5, Section 3, Lemma 3.4. To understand the proof of that lemma, one does not need to read other part of the book - but perhaps some Chapter 1 if one is not familiar with Jacobi field.