We study the problem of determining whether a manufactured disc of certain radius r is within tolerance. More precisely, we present algorithms that, given a set of n probe points on the surface of the manufactured object, compute the thinnest annulus whose outer (or inner, or median) radius is r and that contains all the probe points. Our algorithms run in O(n log n) time.

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Series Lecture Notes in Computer Science
de Berg, M. (Mark), Bose, P, Bremner, D. (David), Ramaswami, S. (Suneeta), & Wilfong, G. (Gordon). (1997). Computing constrained minimum-width annuli of point sets. In Lecture Notes in Computer Science.