Measure Theoretic Versions
In measure theory, Stone & Tukey (1942) proved two more general forms of the ham sandwich theorem. Both versions concern the bisection of n subsets X1, X2, …, Xn of a common set X, where X has a Carathéodory outer measure and each Xi has finite outer measure.
Their first general formulation is as follows: for any suitably restricted real function, there is a point p of the n-sphere Sn such that the surface, dividing X into f(s,x) < 0 and f(s,x) > 0, simultaneously bisects the outer measure of X1, X2, …, Xn. The proof is again a reduction to the Borsuk-Ulam theorem. This theorem generalizes the standard ham sandwich theorem by letting f(s,x) = s0 + s1x1 + ... + snxn.
Their second formulation is as follows: for any n+1 measurable functions f0, f1, …, fn over X that are linearly independent over any subset of X of positive measure, there is a linear combination f = a0f0 + a1f1 + ... + anfn such that the surface f(x) = 0, dividing X into f(x) < 0 and f(x) > 0, simultaneously bisects the outer measure of X1, X2, …, Xn. This theorem generalizes the standard ham sandwich theorem by letting f0(x) = 1 and letting fi(x), for i > 0, be the ith coordinate of x.
Read more about this topic: Ham Sandwich Theorem
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