Local diameters of compact planar sets

Given a compact setS?E² andP ₁ εS, we consider sequencesP _i , 1<i<∞, such that $$h_i : = \left\| {P_i - P_{i + 1} } \right\| = \mathop {\max }\limits_{R \in S} \left\| {P_i - R} \right\|$$ . Callh=limh _i a local diameter ofS. It is realized by a line segmentPQS. In relation toD(S), the ordinary diameter ofS, we determine (with and without certain angular constraints) how smallh can be. For example, ifA, B ε S and the line segmentAB is a diameter ofS, we determine best possible lower bounds for ∥P?Q∥/∥A?B∥ in terms of the smallest angle formed by the lines extendingPQ andAB. Local diameters arise from the problem of computing generalized transfinite diameters. Moreover, information about local diameters provides both upper and lower bounds on the isoperimetric quotient ofS.