David Barr's Four Corners Project begs to be done "better", namely bigger, if only slightly. The earth is not a sphere. We now know the shape of the earth in very high detail. What is the largest volume tetrahedron (not necessarily regular) with all four vertices on land?
Perhaps place vertices on mountains to eke out a slightly larger volume.
The mass of the earth causing gravitational curvature of space by Einstein's general relativity may make volume nontrivial to compute.
Largest volume polyhedra with more vertices (again, not necessarily regular polyhedra)?
Previously mentioned in a comment to this post.
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