Let $\Delta(x,y) = 1,0$ according to whether $(x,y)$ is in some polygon (symmetric with respect to the diagonal axis).
E.g. The convex hull of three points (taken from a paper on dominoes)
$$ \Delta = \bigg\{ r(-1,-1)+s(1,0)+t(0,1): r + s + t = 1\bigg\} $$
So I am looking for eigenvalues and eigenfunctions of this Kernel:
$$ f \mapsto \int_{-1}^1 \Delta(x,\;\cdot\;) f(x)\;dx$$
This occurred to me while reading this paper by Noam Elkies on zeta functions, where he studies the spectrum of the convex hull:
$$ \bigg\{ r(0,0)+s(1,0)+t(0,1): r + s + t = 1\bigg\} $$
In that paper he gets a zeta function by taking the trace of this kernel,
$$ \mathrm{tr}(\Delta^n) = \sum_{i \in \mathbb{Z}} \lambda_i^n $$
For Elkies, $\lambda_k = \frac{1}{4k+1}$. See also the paper by Stanley.
He then interprets the trace $\mathrm{tr}(\Delta^n)$ as the volume of a polytope, which I will address separately.
