Although the Earth is not a perfect sphere, a mathematical object representing its surface (a geoid) is homeomorphic to S^{2}. Therefore, the theorem applies.

Execpt for the fact that, after we've located^{1} the aforementioned two points, and after transforming back to the geoid, the points probably won't be exactly antipodal anymore.

*ariels says "One way to do it is to show that the Earth is star convex around its centre. Define a map *`R^3\{0} -> R^3` by `x -> x/|x|.` Then the map transforms the Earth says to a sphere, while preserving antipodality."

^{1}located, somehow. We were given an existence proof, not a construction.