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A constrained isoperimetric problem

Of all sets lying within a given convex plane set S, which one gives the greatest ratio of area A to perimeter P? This was posed to one of us by E. Bombieri for the case when S is a square. He asserted that T. Bang used an easy upper bound on A/P in an elementary proof of the Prime Number Theorem. After solving this problem from scratch, we learned from M. Silver that Besicovitch (1) had solved a closely related problem, which we call Besicovitch's problem and which will be discussed shortly. Later, we found that the problem of maximizing A/P had occurred to Garvin (4, 5) and that he solved the problem when S is a triangle. Garvin inspired DeMar (2) to consider and solve Besicovitch's problem for the triangle and for polygons. DeMar found that Steiner (8, p. 166) had solved Besicovitch's problem for triangles, but Steiner was only interested in subsets which touched all sides of a polygon (8, p. 168). More recently, J. Wills has referred us to Herz and Kaapke (6) who use Besicovitch's results to solve our problem of maximizing A/P for polygons circumscribed about a circle.