2 See 'update' at end of question

# Characterizing polygonspolyhedron from Brownian particle collisions with a boundary

Please imagine that we have an ordinary 2-sphere, of radius $r_{sphere}$, and some three-dimensional polygon that has all of its points fixed at positions strictly internal to the sphere's surface. Also confined in the sphere is point-like particle (with diffusion constant, $D_{particle}$) undergoing Brownian motion. The surface of the 2-sphere, as well as the surface of the polygon internal to the 2-sphere, are perfect reflecting boundaries for the particle.

Working in discrete time, we track the point-like particle for $N$ finite time units (we'll call them seconds), $(t_1, t_2, ..., t_k, ..., t_N) \in T$. However, during this time the only information we are allowed to record is:

1. If a collision between the probe and the surface of the 2-sphere occurs at a given time point, $t_k$.

And if there is at least one such collision during $t_k$...

1. The coordinates of a collision event on the surface of sphere, randomly selected from all collisions that occur during $t_k$.

Beyond, perhaps, the volume of the polygon in the sphere (and I'm not entirely sure this is learnable), how (if at all possible) can we use the information specified above to characterize the polygon in some additional manner?

Update - If we apply a further restriction that the polyhedron is convex, at the limit of large $N$ will there be enough information from the collisions to reconstruct the convex polyhedra?

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# Characterizing polygons from Brownian particle collisions with a boundary

Please imagine that we have an ordinary 2-sphere, of radius $r_{sphere}$, and some three-dimensional polygon that has all of its points fixed at positions strictly internal to the sphere's surface. Also confined in the sphere is point-like particle (with diffusion constant, $D_{particle}$) undergoing Brownian motion. The surface of the 2-sphere, as well as the surface of the polygon internal to the 2-sphere, are perfect reflecting boundaries for the particle.

Working in discrete time, we track the point-like particle for $N$ finite time units (we'll call them seconds), $(t_1, t_2, ..., t_k, ..., t_N) \in T$. However, during this time the only information we are allowed to record is:

1. If a collision between the probe and the surface of the 2-sphere occurs at a given time point, $t_k$.

And if there is at least one such collision during $t_k$...

1. The coordinates of a collision event on the surface of sphere, randomly selected from all collisions that occur during $t_k$.

Beyond, perhaps, the volume of the polygon in the sphere (and I'm not entirely sure this is learnable), how (if at all possible) can we use the information specified above to characterize the polygon in some additional manner?