Somewhere in a two dimensional convex bulk of particles (pic related) on a random position a reaction takes place and a particle is sent out in a random direction with a constant velocity $v$.
What is the average distance such a particle travels until it leaves the bulk?
Might be codable if one puts a grid over the planes and weightens with a yes/no function if the stating position is within the bulk. Then I'd cut that object with n corners (8 in the pic) into pizza slices and do some geometry to compute the distances in all directions and integrate over all of these and all staring point. I really wonder if there is a good way to do this on a piece of paper, the problem being how to parametrize the points which are inside and not counting these outside.
Monte-Carlo computations for regular polygons would be an interesting semi-solution too.
Edit: Not that it matters much, but the question is motivated by the question Mean free path of UV photon and is part of me wondering about the escape route for particle entering a bunch of mass, and there specifically on the the dependence on the two length characteristica cloud dimensions and mean free path derived by the clouds constitution. Given that the direction of a particle after a collision is random and so it will probably have to make a detour through the cloud, what is the relation between the average cloud radius to the mean free patch such that the particle is able to leave after only one reaction. Because a collision is almost a reset, I have the suspicion that the escape time only falls slowly with the number of collisions a particles had to endure. The simulation of this might be a more standard question, i.e. starting at a random position and choosing a random direction after the mean free path traveled, how manny mean free paths does it take for a particle to escape a random polygon with characteristic measure being some multiple of the mean free path.