In 2D perimeter(P) of a convex set around origin may be written as $P=1/2 \int m(\theta) d\theta$. Where $m(\theta)$ is the diameter of the set in the $\theta$ direction. This is related to Cauchy-Crofton formula. The question is if anything similar known at higher dimensions ?



Surface area of a convex set
A good reference is Klain and Rota's little book *Introduction to Geometric Probability*, especially Section 5.5. 


Here's the formula. The surface area of a compact convex subset $K$ of $\mathbb{R}^n$ is


$$
\frac{1}{\omega\_{n - 1}} \int\_{S^{n-1}} Vol\_{n-1}(\pi\_{\theta^{\bot}} K) d\theta.
$$


Here $\omega\_{n - 1}$ is the volume of the unit Euclidean $(n - 1)$-ball, $\theta^\bot$ is the linear subspace of $\mathbb{R}^n$ orthogonal to the point $\theta$ of $S^{n - 1}$, and $\pi\_{\theta^{\bot}}$ is orthogonal projection onto that subspace; also, $Vol\_{n - 1}$ is Lebesgue measure on $\mathbb{R}^{n - 1}$. 


As you may know, this is a special case of the more general "Crofton formula" for the intrinsic volumes. In the surface area formula, we're effectively integrating over the space of all $(n - 1)$-dimensional linear subspaces of $\mathbb{R}^n$; for the general Crofton formula, we integrate over the space of all $k$-dimensional linear subspaces of $\mathbb{R}^n$, for some fixed $k$. This gives the formula for the $k$-dimensional intrinsic volume. All this is nicely explained in Klain and Rota.


