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I'm trying to use Crofton's formula to prove that $$\int \kappa\,ds \geq 2\pi$$ for all closed regular curves.

What I note is that Crofton's formula says if $C$ is the image of a regular curve on $S^2$ of length $l$, then the measure of the set of oriented circles intersected $C$, with multiplicity, is $4l$.

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Let $\alpha(s)$ be a unit speed, simple, closed curve. Then let $\gamma(s)=\alpha^\prime(s)$. Thus, $\gamma$ is a closed curve on the unit sphere. So, we would like to apply Crofton's formula to $\gamma$.

Crofton's formula says that $$4L = {\int \! \!\int}_{S^2} n(W)dS$$

where $n(W)$ counts the number of times that the curve $\gamma$ intersects the equator with $W$ taken to be the north pole, and $L$ is the length of the curve. So, $$L = \int \| \gamma^\prime \|ds$$

But, $\| \gamma^\prime \|$ is just $\kappa$, the curvature of $\alpha$. Thus, we end up with $$\int \kappa ds = \frac{1}{4} {\int \! \!\int}_{S^2} n(W)dS$$

Suppose we knew that $n(W)\ge 2$ for all points $W$ on the sphere for this curve. Then we would get, $$\int \kappa ds \ge \frac{2}{4} {\int \! \!\int}_{S^2} dS$$ $$\int \kappa ds \ge \frac{1}{2}(4 \pi)$$ $$\int \kappa ds \ge 2\pi$$

Which is exactly what we want to show! Can you prove what I claim about $n(W)$?

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