Summary.
One of Paul Erdős’s many continuing interests is distances between points in finite sets. We focus here on conjectures and results on intervertex distances in convex polygons in the Euclidean plane. Two conjectures are highlighted. Let t(x) be the number of different distances from vertex x to the other vertices of a convex polygon C, let \(T(C) = \Sigma t(x)\), and take \(T_{n} =\min \{ T(C) : C\mbox{ has $n$ vertices}\}\). The first conjecture is \(T_{n} = \left ({ n \atop 2} \right )\). The second says that if \(T(C) = \left ({ n \atop 2} \right )\) for a convex n-gon, then the n-gon is regular if n is odd, and is what we refer to as bi-regular if n is even. The conjectures are confirmed for small n.
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References
E. Altman, On a problem of P. Erdős, Amer. Math. Monthly 70 (1963) 148–157.
H. Edelsbrunner and P. Hajnal, A lower bound on the number of unit distances between the vertices of a convex polygon, J. Combin. Theory A 56 (1991) 312–316.
P. Erdős, On sets of distances of n points, Amer. Math. Monthly 53 (1946) 248–250.
P. Erdős and P. Fishburn, Multiplicities of interpoint distances in finite planar sets, Discrete Appl. Math. (to appear).
P. Erdős and P. Fishburn, Intervertex distances in convex polygons, Discrete Appl. Math. (to appear).
P. Erdős and P. Fishburn, A postscript on distances in convex n-gons, Discrete Comput. Geom. 11 (1994) 111–117.
P. Erdős and L. Moser, Problem 11, Canad. Math. Bull. 2 (1959) 43.
P. Fishburn, Convex polygons with few intervertex distances, DIMACS report 92–18 (April 1992), AT&T Bell Laboratories, Murray Hill, NJ.
P. Fishburn, Convex polygons with few vertices, DIMACS report 92–17 (April 1992), AT&T Bell Laboratories, Murray Hill, NJ.
P. C. Fishburn and J. A. Reeds, Unit distances between vertices of a convex polygon, Comput. Geom.: Theory and Appls. 2 (1992) 81–91.
Z. Füredi, The maximum number of unit distances in a convex n-gon, J. Combin. Theory A 55 (1990) 316–320.
L. Moser, On the different distances determined by n points, Amer. Math. Monthly 59 (1952) 85–91.
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Fishburn, P. (2013). Distances in Convex Polygons. In: Graham, R., Nešetřil, J., Butler, S. (eds) The Mathematics of Paul Erdős I. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7258-2_30
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