Diameter bounded from above is usually needed in the finiteness theorem or other convergence theorems in Riemannian Geometry. Let $M^n$ be a closed manifold and {$g\_i$} be a family of smooth Riemannian metrics on it with $Inj\_{g\_i}\geq \alpha>0$ and $Vol\_{g\_i}\leq \beta$. Are those two conditions enough to imply $Diam\_{g\_i}\leq \gamma$? If so, does the $\gamma$ depend only on $\alpha,\beta$ and $n$?



Can the diameter be controled by the injectivity radius and the volume?
By [Croke, Some isoperimetric inequalities and eigenvalue estimates](http://www.numdam.org/article/ASENS_1980_4_13_4_419_0.pdf), Proposition 14, on an $n$-dimensional Riemannian manifold with injectivity radius $\alpha$, a ball of radius $\alpha/2$ has volume at least $C\alpha^n/ (2n)^n$ for a constant $C$. Specifically, by their theorem 11, $C=2^{n-1}V\_{n-1}^{n\phantom{1}} / V\_{n\phantom{1}}^{n-1}$, where $V\_n$ is the volume of the $n$-dimensional sphere.


Let $x\_1,\dots,x\_m$ be a maximal collection of points such that the balls of radius $\alpha/2$ centered at $x\_1,\dots,x\_m$ do not overlap. Then clearly $$m\frac{ C\alpha^n }{ (2n)^n} \leq \beta.$$ By maximality, the balls of radius $\alpha$ centered at $x\_1,\dots, x\_m$ cover $M$. Hence because $M^n$ is connected, the graph with vertices $x\_1,\dots,x\_m$ and edges connecting $x\_i$ to $x\_j$ if the distance from $x\_i$ to $x\_j$ is at most $2\alpha$ is connected and thus has diameter at most $m-1$.


Thus the diameter is at most $$\alpha + (m-1) 2\alpha+ \alpha = 2m\alpha \leq 2 \beta \frac{ (2n)^n}{ C \alpha^n} \alpha = \frac{ \beta}{\alpha^{n-1} } \frac{2^{n+1} n^n} {C} $$ as for the distance between $x$ and $y$, it takes a distance of $\alpha$ to get from $x$ to one of the $x\_i$, then $(m-1)$ distances of $2\alpha$ to travel to an appropriate $x\_j$, and then $\alpha$ to get from $x\_j$ to $y$.


This is sharp up to a constant depending on $n$, as demonstrated by a suitable long thin torus.


