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Fair Partitioning by Straight Lines
A pizza is a pair of planar convex bodies A⊆B, where B represents the dough and A the topping of the pizza. A partition of a pizza by straight lines is a succession of double operations: a cut by a full straight line, followed by a Euclidean move of one of the resulting pieces; then the procedure is repeated. The final partition is said to be fair if each resulting slice has the same amount of A and the same amount of B. This note proves that, given an integer n≥2n≥2, there exists a fair partition by straight lines of any pizza (A, B) into n parts if and only if n is even. The proof uses the following result: For any planar convex bodies A, B with A⊆B, and any α∈]0,1/2[, there exists an αα-section of A which is a β-section of B for some β≥α. (An α-section of A is a straight line cutting A into two parts, one of which has area α|A|). The question remains open if the word “planar” is dropped.