# dg.differential geometry – Optimal lower bound on the volume of balls under a Sobolev inequality

Let $$M$$ be a complete non-compact $$n$$-dimensional ($$n geq 3$$)
Riemannian manifold with volume element
$$dv$$ such that, for every smooth compactly supported function $$f : M to mathbb {R}$$,
$$bigg ( int_M |f|^{frac {2n}{n-2}} dvbigg)^{frac {n-2}{n}} , leq , C int_M |nabla f|^2 dv$$
where $$C >0$$ is the optimal constant of this Sobolev inequality in the Euclidean case
$$M = mathbb{R}^n$$. Is it true that
$$mathrm {Vol} (B(x,r) ) , geq , mathrm{V}(r)$$
where $$B(x,r)$$, $$x in M$$, is a ball of radius $$r >0$$ in $$M$$ and
$$mathrm{V}(r)$$ is the volume of a ball of radius $$r$$ in $$mathbb{R}^n$$?