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Compute mean tetrahedron volume for starred 3D polyhedra

Determine the exact mean volume v_3^{(1)}(P_3) of a random tetrahedron formed by four independent uniform points in each of the starred polyhedra listed in Table 2 (triakis tetrahedron, tetrakis hexahedron, and truncated octahedron), with each body normalized to unit volume.

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Background

In three dimensions the authors computed v_3{(1)}(P_3) for a broad set of polyhedra, but note that several marked with an asterisk remain uncomputed in the present manuscript.

They indicate that section integrals for these cases are currently available only in some genealogies and anticipate providing the missing evaluations in a future update.

References

To be honest with the reader, the polyhedra indicated by $*$ have not been computed yet (section integrals are availible only in some particular genealogies), but they will surely appear in an updated version of this paper.

— On Random Simplex Picking Beyond the Blashke Problem (2412.07952 - Beck, 10 Dec 2024) in Section: Three dimensions, after Table “Polyhedra for which we considered v_3^{(1)}(K_3)”