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Consider a convex set $K$ and and a Poisson random polygon $P$ inside of it. So the polygon $P$ is generated by using a poisson point process to generate the points and then taking the convex hull. Now let $P_{K}(\theta)$ be a vertex of $P$ which has an oriented tangent line at angle $\theta$. What is the probability distribution of $P_{K}(\theta)$?

It seems that if you fix $\theta$ then there could be some vertices of $P$ that do not have an oriented tangent line at angle $\theta$. So $P_{K}(\theta)$ is unique for any given $\theta$?

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