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I wish to prove the following assertion:

The measure of the set of lines that meet a convex closed curve $C$ (without multiplicites) is equal to the length of $C$.

I know this is an application of Cauchy-Crofton theomrem, saying that:

Let $C$ be a regular plane curve with length $L$. The measure of the set of straight lines (counted with multiplities) which meet $C$ is 2$L$.

For the convex closed curve $C$, a line is either tangent to it with only one interestion with $C$ or there are exactly 2 intersections. Then, in the problem, we neglect the multiplicities, then we have the result of $L$.

I feel this is quite a naive argument. Could anyone formalize it? Thanks in advance.

Scorpio19891119
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1 Answers1

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The set of lines that are tangent to a regular curve, $C$, has measure zero, so the tangent lines can be ignored. For the secant lines, each one intersects the curve twice. So the measure, with multiplicities, is twice the measure you get if you count each secant line only once. Cauchy Crofton does the rest.

The only subtlety here is the measure-zero argument. It depends on regularity, and probably some application of Sard's Theorem. But if you think of line-space as parameterized by "closest-point-to-origin, angle-to-x-axis", then the set of tangent lines constitutes a parameterized curve in this space; one that intersects each angle = const coordinate exactly twice. That's probably enough to prove its measure is zero.

John Hughes
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