Stromquist–Woodall theorem
The Stromquist–Woodall theorem is a theorem in fair division and measure theory. Informally, it says that, for any cake, for any "n" people with different tastes, and for any fraction "r", there exists a subset of the cake that all people value at exactly a fraction "r" of the total cake value. The theorem is about a circular 1-dimensional cake (a "pie"). Formally, it can be described as the interval [0,1] in which the two endpoints are identified. There are "n" continuous measures over the cake: formula_1; each measure represents the valuations of a different  person over subsets of the cake. The theorem says that, for every weight formula_2, there is a subset formula_3, which is a union of at most formula_4 intervals, which all people value at exactly formula_5: Proof sketch. formula_7 be the subset of all weights for which the theorem is true. Then: From 1-4, it follows that formula_21. In other words, the theorem is valid for "every" possible weight. Tightness proof. Stromquist and Woodall prove that the number formula_4 is tight if the weight formula_5 is either irrational, or rational with a reduced fraction formula_61 such that formula_62. Stromquist–Woodall