Local diameters of compact planar sets (Q1911994)

Given a compact set \(S \subset R^2\) and \(P_1 \in S\), the sequence \(P_i\) \((i = 1, 2, \dots)\) is defined by \(|P_i - P_{i + 1} |= \max_{R \in S} |P_i - R |\). The paper under review investigates the behaviour of the sequence of Euclidean distances \(|P_i - P_{i + 1} |\) and their limit. Among other things it is proved that \(|P_1 - P_2 |\geq {1 \over 2} D(S)\) and \(|P_i - P_{i + 1} |\geq {1 \over \sqrt 3} D(S)\) for \(i \geq 2\), where \(D(S)\) denotes the diameter of \(S\).