Recently it has been shown that the heuristic Rosenfeld functional derives
from the virial expansion for particles which overlap in one center. Here, we
generalize this approach to any number of intersections. Starting from the
virial expansion in Ree-Hoover diagrams, it is shown in the first part that
each intersection pattern defines exactly one infinite class of diagrams.
Determining their automorphism groups, we sum over all its elements and derive
a generic functional. The second part proves that this functional factorizes
into a convolute of integral kernels for each intersection center. We derive
this kernel for N dimensional particles in the N dimensional, flat Euclidean
space. The third part focuses on three dimensions and determines the
functionals for up to four intersection centers, comparing the leading order to
Rosenfeld's result. We close by proving a generalized form of the Blaschke,
Santalo, Chern equation of integral geometry.Comment: 2 figure
