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min of max (PH_i) when H_iH_j <= 1

Source: 1964 Swedish Mathematical Competition p4

March 21, 2021
minmaxcombinatorial geometrygeometryinequalitiesGeometric Inequalities

Problem Statement

Points H1,H2,...,HnH_1, H_2, ... , H_n are arranged in the plane so that each distance HiHj≤1H_iH_j \le 1. The point PP is chosen to minimise max⁡(PHi)\max (PH_i). Find the largest possible value of max⁡(PHi)\max (PH_i) for n=3n = 3. Find the best upper bound you can for n=4n = 4.
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