On the area sum of a convex polygon and its polar reciprocal.

Florian, August

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On the area sum of a convex set and its polar reciprocal.

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Let B be a convex body in R, with piecewise smooth boundary and let ^χ denote the Fourier transform of its characteristic function. In this paper we determine the admissible decays of the spherical L averages of ^χ and we relate our analysis to a problem in the geometry of convex sets. As an application we obtain sharp results on the average number of integer lattice points in large bodies randomly positioned in the plane.