The ratio of the surface volume to hypervolume of an $n$-sphere is $\frac{r}{(n+1)}$ . Is there a way to prove that this is the highest ratio possible without appealing to the fact that $n$-spheres maximize this quantity?
This falls right out of the volume and surface area formulas
$$ V_n(r) = \frac{\pi^{n/2}r^n}{\Gamma(\frac{n}{2} + 1)} $$
dividing $V_n$ by its derivative with respect to $r$ gives us $n+1$, or the dimension of the space we're embedding the sphere in.
For an arbitrary non-self-intersecting $n$-manifold enclosing finite volume in $\mathbb{R}^{(n+1)}$ , the definition of the surface area and volume are straightforward. The radius is a little bit tricker, but I think it makes sense to use $\frac{d}{2}$ with $d$ being the maximum distance between any two points on the surface. This notional radius is only there to "normalize" the ratio based on the size of the shape in a way that keeps the resulting expression friendly.
Is there a way of proving that the maximum attainable value for $\frac{V_n}{\text{SA}\cdot r}$ is $\frac{1}{n+1}$ without first demonstrating that the most efficient shape is an $n$-sphere?