Claims

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1952_02_01_moser: Moser's 1952 argument gives every convex n-gon a vertex with at least ⌈n/3⌉ distinct distances to the others, which reaches ⌊n/2⌋ for n = 3, 4, 5 and 7; Amer. Math. Monthly 1952.

1994_01_01_erdos_fishburn: Erdős and Fishburn prove that every convex n-gon, n ≥ 4, has a vertex followed by ⌊n/3⌋+1 successively farther vertices, so that many distinct distances, which reaches ⌊n/2⌋ for n = 4 to 7 and 9; DCG 1994.

2006_09_29_dumitrescu: Dumitrescu proves that every convex n-gon has a vertex with at least ⌈(13n-6)/36⌉ distinct distances to the others, which reaches ⌊n/2⌋ for n = 3, 4, 5, 7 and 9; Discrete Comput. Geom. 2006.

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