A postscript on distances in convexn-gons
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Letg(n) be the largest integerk such that every convex polygon withn vertices and sides has a vertexx such that the nextk vertices clockwise fromx, or the nextk vertices counterclockwise fromx, are successively farther fromx. We prove thatg(n)=[n/3]+1 forn≥4. An example givesg(n)≤[n/3]+1, and an extension of a 1952 construction of Leo Moser for a related planar problem shows thatg(n)≥[n/3]+1.
KeywordsDiscrete Comput Geom Convex Polygon Adjacent Vertex Regular Polygon Interior Angle
- 3.P. Erdős and P. C. Fishburn, Multiplicities of interpoint distances in finite planar sets,Discrete Appl. Math., in press.Google Scholar