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This paper, about a fluid-like system of spatially confined particles,reveals the analytic structure for both, the canonical and grand canonicalpartition functions. The studied system is inhomogeneously distributed in aregion whose boundary is made by planar faces without any particular symmetry.This type of geometrical body in the $d$-dimensional space is a polytope. Thepresented result in the case of $d=3$ gives the conditions under which thepartition function is a polynomial in the volume, surface area, and edgeslength of the confinement vessel. Equivalent results for the cases $d=1,2$ arealso obtained. Expressions for the coefficients of each monomial are explicitlygiven using the cluster integral theory. Furthermore, the consequences of thepolynomial shape of the partition function on the thermodynamic properties ofthe system, away from the so-called thermodynamic limit, is studied. Someresults are generalized to the $d$-dimensional case. The theoretical toolsutilized to analyze the structure of the partition functions are largely basedon integral geometry.

Statistical mechanics of fluids confined by polytopes: The hidden geometry of the cluster integrals