A converse to the discrete four-vertex theorem
Tabachnikov asked whether every cyclic sequence of positive numbers with at least two local maxima and two local minima is the sequence of radii of the circles through consecutive vertex triples of a convex polygon whose circumcenters lie inside the corresponding vertex angles. We prove this for every n ≥ 4, with extrema counted as plateaus. This is the sharp form: by Dahlberg’s discrete four-vertex theorem the condition is also necessary for nonconstant sequences. Locally near every nonconstant admissible sequence the polygon can be chosen to depend continuously (indeed C1) on the sequence,…
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