Inequalities for integral means over symmetric sets |
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Authors: | Cristina Draghici |
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Institution: | Department of Mathematical Sciences, Aalborg University, Fr. Bajers Vej 7G, DK-9220 Aalborg East, Denmark |
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Abstract: | We prove that the integral of n functions over a symmetric set L in Rn, with additional properties, increases when the functions are replaced by their symmetric decreasing rearrangements. The result is known when L is a centrally symmetric convex set, and our result extends it to nonconvex sets. We deduce as consequences, inequalities for the average of a function whose level sets are of the same type as L, over measurable sets in Rn. The average of such a function on E is maximized by the average over the symmetric set E*. |
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Keywords: | Symmetrization Rearrangement Integral inequality |
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