Abstract
For a region X in the plane, we denote by area(X) the area of X and by ℓ (∂ (X)) the length of the boundary of X . Let S be a convex set in the plane, let n ≥ 2 be an integer, and let α 1 , α 2 , . . . ,α n be positive real numbers such that α 1 +α 2 + ⋅ ⋅ ⋅ +α n =1 and 0< α i ≤ 1/2 for all 1 ≤ i ≤ n . Then we shall show that S can be partitioned into n disjoint convex subsets T 1 , T 2 , . . . ,T n so that each T i satisfies the following three conditions: (i) area(T i )=α i × area(S) ; (ii) ℓ (T i ∩ ∂ (S))= α i × ℓ (∂ (S)) ; and (iii) T i ∩ ∂ (S) consists of exactly one continuous curve.
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Kaneko, Kano Perfect Partitions of Convex Sets in the Plane. Discrete Comput Geom 28, 211–222 (2002). https://doi.org/10.1007/s00454-002-2808-2
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DOI: https://doi.org/10.1007/s00454-002-2808-2