Physical Significance of $8\pi r$ (the second derivative of the volume of a sphere of radius $r$)

$8\pi r = 4\pi \times \ell = 2\times H (4\pi r^2)$ where $\ell = 2r$ is the mean width for a ball with radius $r$ and $H = \frac1r$ is the mean curvature for corresponding sphere.

Given any convex body $K$ in $\mathbb{R}^3$ and unit vector $n \in S^2$, let $w(K,n)$ be the width of shadow when one project $K$ orthogonally to the line $\{ \lambda n : \lambda \in \mathbb{R} \}$, the mean width of $K$ is the angular average of $w(K,n)$ for $n$ varies over $S^2$:

$$\ell(K) \stackrel{def}{=} \frac{1}{4\pi} \int_{S^2} w(K,n)$$

For any $\epsilon > 0$, let $B_\epsilon = \{ x \in \mathbb{R}^3 : |x| \le \epsilon \}$ be the sphere centered at origin with radius $\epsilon$.
If one form the Minkowski sum of $K$ with $B_\epsilon$, i.e.

$$K_\epsilon = \{ x + y : x \in K, y \in B_\epsilon \}$$

The volume of $K_\epsilon$ depends on $\epsilon$ polynomially,

$$\verb/Vol/(K_\epsilon) = \verb/Vol/(K) + \verb/Area/(K)\epsilon + 2\pi\ell(K)\epsilon^2 + \frac{4\pi}{3} \epsilon^3$$

This means in general, we have

$$\left.\frac{d^2}{d\epsilon^2}\verb/Vol/(K_\epsilon) \right|_{\epsilon=0}= 4\pi \ell(K)$$

In particular, for a sphere with radius $r$, we have

$$(B_r)_\epsilon = B_{r+\epsilon} \quad\implies\quad \frac{d^2}{dr^2}\verb/Vol/(B_r) = \frac{d^2}{d\epsilon^2} \verb/Vol/((B_r)_\epsilon) = 4\pi \ell(B_r) = 4\pi(2r) = 8\pi r$$

When $K$ is smooth enough, the mean width of $K$ is related to the average of mean curvature $H$ over the surface bounding $K$. In fact

$$\ell(K) = \frac{1}{2\pi}\int_{\partial K}H \quad\implies\quad \left.\frac{d^2}{d\epsilon^2}\verb/Vol/(K_\epsilon)\right|_{\epsilon=0} = 2\int_{\partial K}H$$ For a sphere with radius $r$, the expression $8\pi r$ can be decomposed as

$$8 \pi r = 2 \times \frac1r(4\pi r^2)$$

and the RHS can be interpreted as twice the integral of mean curvature $H = \frac1r$ over a surface of area $4\pi r^2$.