Abstract
The (distance) k-sector is a generalization of the concept of bisectors proposed by Asano, Matoušek and Tokuyama. We prove the uniqueness of the 4-sector of two points in the Euclidean plane. Despite the simplicity of the unique 4-sector (which consists of a line and two parabolas), our proof is quite non-trivial. We begin by establishing uniqueness in a small region of the plane, which we show may be perpetually expanded afterward.
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Fraser, R., He, M., Kawamura, A., López-Ortiz, A., Munro, J.I., Nicholson, P.K. (2013). The Distance 4-Sector of Two Points Is Unique. In: Cai, L., Cheng, SW., Lam, TW. (eds) Algorithms and Computation. ISAAC 2013. Lecture Notes in Computer Science, vol 8283. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45030-3_57
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DOI: https://doi.org/10.1007/978-3-642-45030-3_57
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