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What if it's not exactly a sphere and the measures are not exactly continuous?

The measures need to be continuous but it doesn't need to be a sphere. It can be a warped or stretched sphere as long as the space is also continuous.

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What if the Earth is flat or toroidal?

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Flat definitely wouldn't work. Using the walking around the equator logic... well that logic breaks if there's no continuous loop.

Not sure about toroidal.

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My intuition says you'd maintain equivalence of one measure, but maybe not two.

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