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 CauchyCrofton formula. The question is if anything similar known at higher dimensions ?

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 $$ \frac{1}{\omega_{n  1}} \int_{S^{n1}} Vol_{n1}(\pi_{\theta^{\bot}} K) d\theta. $$ 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. 

