I have no idea what your objection means.

In no particular order: take the discrete Fourier transform; read the faq; ask on math.stackexchange.com

Under the conditions you list, and with $\pi$ given, I think that $P$ can be described by specifying for each unordered pair $\lbrace i,j\rbrace$ a transition rate $P_{ij}$. The reverse rate $P_{ji}$ is determined by detailed balanced. But for each pair one rate is completely free and independent of all other rates so long as you satisfy ergodicity ($(P^n)_{ij}>0$ for some $n$).

Neil Sloane's home page neilsloane.com has links to some of his other useful databases cataloging, for example, lattices, spherical codes, Lennard Jones cluster, Hadamard matrices, and so on.

You could ask that all coordinates are algebraically independent.

It seems from numerics that it is indeed the case that $f$ is weakly unimodal. So this is a valuable example to keep in mind. However, as I have pointed out, I am interested in the cases where $f(\theta)$ is not constant on any interval. Particularly, I want to eliminate the cases that $K$ or $L$ are circular disks and the cases where $R_\theta(L)\subset K$ for some $\theta$.

Yes, you're right. I actually should have written "increasing" instead of "weakly increasing". I will edit accordingly.

This is basically the answer given on m.se, so I wonder what Paul found unsatisfying about the answer there.

Here is the question on m.se: math.stackexchange.com/questions/316226

I wonder what you mean by the perimeter? Is this the same as the surface area in $R^3$? If so, then what about the catenoid? Its convex hull has a larger surface area (the corners can be smoothed out to be $C^1$).

For arbitrary $p_i$, calculating the value of $\mu(P)$ for some $P$ is equivalent to the subset sum problem, an NP-Complete problem.

More from Wikipedia: en.wikipedia.org/wiki/Disk_covering_problem