This paper extends existing results on irregularities of distribution and geometric inequalities for convex bodies.
W. Schmidt, H. Montgomery, and J. Beck proved a result on irregularities of distribution with respect to d-dimensional balls. In this paper, we extend their result to any d-dimensional convex body with a smooth boundary and finite order of contact. As an intermediate step, we prove a geometric inequality for the Fourier transform of the characteristic function of a convex body.
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Brandolini et al. (2025) studied this question.
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