In this paper we relate the minimal annulus of a planar convex body with its inradius, obtaining all the upper and lower bounds, in terms of these quantities, for the classic geometric measures associated with the set: area, perimeter, diameter, minimal width and circumradius. We prove the optimal inequalities for each one of those problems, determining also its corresponding extremal sets.
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María A. Hernández Cifre, Pedro J. Herrero Piñeyro, Optimizing geometric measures for fixed minimal annulus and inradius. Rev. Mat. Iberoam. 23 (2007), no. 3, pp. 953–971DOI 10.4171/RMI/520