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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 ?

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There are Cauchy-Crofton formulas in all dimensions. The constants in front of the integrals are certain canonical measures on Grassmannians, and you are integrating over spaces of affine lines, and affine hyperplanes, etc, depending on the application. But there are versions that apply to your situation -- using spaces of affine lines. – Ryan Budney Apr 20 '13 at 8:01
I know some people use these generalized Cauchy-Crofton theorems in their work but off the top of my head I don't know a great reference for them. They come in a wide variety of flavours. They're not very hard to prove -- kind of a fun exercise in the basic geometry of the canonical bundles over Grassmannians together with the Fubini theorem and various change-of-variables theorems for integration. – Ryan Budney Apr 20 '13 at 8:08
Thanks, it would be good to get a reference to look at. I actually need the formula. – pallab1234 Apr 20 '13 at 11:54
up vote 2 down vote accepted

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.

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