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From Wikipedia: The problem of whether a given equation holds in every Heyting algebra was shown to be decidable by S. Kripke in 1965.[1] The precise computational complexity of the problem was established by R. Statman in 1979, who showed it was PSPACE-complete[7] and hence at least as hard as deciding equations of Boolean algebra (shown NP-complete in 1971 by S. Cook)[8] and conjectured to be considerably harder. en.wikipedia.org/wiki/Heyting_algebra
By symmetry the expectation would be zero winding, but the interesting question would be how the variance grows with time. Here is the surface on which I was suggesting to solve the diffusion equation: en.wikipedia.org/wiki/File:Riemann_surface_log.jpg
Can't you equivalently consider a single random walk's winding number about the origin? I think you must put some barrier at the origin, because (in the plane) I think it will hit the origin with non-zero probability. Once you figured out the right set-up, you can convert the discrete random walk into a diffusion equation on the Riemann surface for $\log z$, and if you can solve the diffusion equation you will have your answer.
Incidentally, I am writing this from a hotel in Palo Alto where I am staying for the duration of a workshop in the American Institute of Mathematics on "Sections of convex bodies" starting tomorrow: aimath.org/ARCC/workshops/sectionsconvex.html
Also related: there are non-spherical convex bodies such that the maximal cross sectional area in every direction is the same. See mathoverflow.net/questions/70391/…
Among bodies of revolution with constant brightness, the Blaschke-Fiery body is of minimal volume. That's the body whose surface area measure is constant for $0<z<a_n$ or $z<-a_n$ and zero otherwise, where $a_n = 1/2$ in $n=3$. See P. Gronchi, "Bodies of constant brightness", Archiv der Mathematik 70, pp. 489-498, 1998.
Let $S_K(x,y,z) = 1 + a P_3(z)$, where $P_3(z)=(5z^3-3z)/2$ is the Legendre Polynomial, and $a$ is small enough so that $S_K$ is non-negative. By the existence theorem for the Minkowski problem, this is the surface area measure of some convex body.
Do you count +1 when the extension of the normal sweeps across p in one direction and -1 when it sweeps across p in the other direction, or do you count +1 for both?
As for efficiency of calculating this area at each step, it seems it should be too inefficient. If for some reason it is, I can think of many ways of approximating it cheaply.
For the circle circumscribing the region, this is also easy if you think about the three cases above. Obviously, this is not applicable for regions that abut the boundary of $P$.
In the generic case, your region will be bounded by three circles (possibly of zero curvature) that are tangent to each other. Calculating the area of this region is easy (think of the triangle formed by the centers minus the three circular sectors). Other cases that arise, but are not much harder: near corners of $P$ you will have a region bounded by two lines and a circle tangent to them; you will unavoidably have some regions bounded by a cycle of four tangent circles.
