Perfect partitions of convex sets in the plane (Q1864188)

Let \(S\) be a convex set in the plane, and let \(\alpha _{1},\dots ,\alpha _{n}\) be real numbers such that \(\alpha _{1}+\dots +\alpha _{n}=1\) and \(0<\alpha _{i}\leq 1/2\) for all \(i\in \{1,\dots ,n\}\). The main theorem of this paper says that \(S\) can be partitioned into \(n\) convex subsets \(T_{1},\dots ,T_{n}\) such that each \(T_{i}\) satisfies the following conditions: (i) \(\lim\text{area}(T_{i})=\alpha _{1}\times \lim\text{area}(S)\), (ii) \(\ell (T_{i}\cap \partial S)=\alpha _{i}\times \ell (\partial S)\), and (iii) \(T_{i}\cap \partial S\) consists of exactly one continuous curve. (Here \(\partial S\) denotes the boundary of \(S\), and \(\ell (C)\) denotes the length of the curve \(C\).) There is also a discussion of the relation of this theorem to the problem of partitioning a finite set of red and blue points in the plane into \(n\) subsets with pairwise disjoint convex hulls such that each subset contains the ``right'' number of red and blue points.