Long time no see. : comments [Known Plaintext]

By "body", I assume you mean something with volume. Call them V1, V2, V3. For each V, define a center P according to some property of uniqueness -- center of mass, whatever, so that each V maps onto a P.

You now have three central points P which define a plane which bisects each V.

I maintain the bisecting plane is trivial to construct in the first place, assuming sensible choice of the function mapping V => P. :)

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For "body", read "continuous volume", yes.

It's not really a question of constructing the bisecting plane, it is a question of "considering all points in all planes that exactly halve a body, is there, for all bodies, at least one point that is in all the halving planes".

I have not yet managed to construct a counterexample (but there are bodies for which said point is actually outside the body, like, say, a torus).

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