The Topping's diameter estimates is
Let $M$ be an $n$-dimensional closed, connected manifold smoothly immersed in $\mathbb R^N$, where $N\ge n+1$. Then the intrinsic diameter and the mean curvature $H$ of $M$ are related by $$ \operatorname{diam}(M)\le C(n)\int_M |H|^{n-1} $$
I feel there should be some lower estimation of $\operatorname{diam}(M)$ has liking form, since obviously, we have $$ \frac{1}{\max\limits_{M} |A|} \le \operatorname{diam}(M) $$ although it is very rough, it means there should be some lower estimation. What theory describe this ?
Let $S_r$ be the smallest sphere of radius $r$ which contains $M$. Then the sphere touches $M$ at $x\in M$. At this point we have $$ |A(x)| \ge |A_r(x)|= \frac{n}{r},$$ (here $A_r$ is the second fundamental form of $S_r$). Thus $$ |A(x)| r \ge n\Rightarrow \max|A|\operatorname{diam}(M) \ge 2n.$$