Equality conditions for the fractional superadditive volume inequalities
Meyer, Mark
Meyer, Mark
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2024
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Article
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52A40,Brunn-Minkowski,Convex hull,Lebesgue measure,Sumsets
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Meyer, M. Equality conditions for the fractional superadditive volume inequalities. (2024). Discrete and Computational Geometry. DOI: 10.1007/s00454-024-00672-8
Abstract
While studying set function properties of Lebesgue measure, F. Barthe and M. Madiman proved that Lebesgue measure is fractionally superadditive on compact sets in R$^n$. In doing this they proved a fractional generalization of the Brunn–Minkowski–Lyusternik (BML) inequality in dimension n=1. In this paper we will prove the equality conditions for the fractional superadditive volume inequalites for any dimension. The non-trivial equality conditions are as follows. In the one-dimensional case we will show that for a fractional partition (G,β) and nonempty sets A1,⋯,A$_m$⊆R, equality holds iff for each S∈G, the set ∑$_{i∈S}$A$_i$ is an interval. In the case of dimension n≥2 we will show that equality can hold if and only if the set ∑i=1$^m$A$_i$ has measure 0. © The Author(s) 2024.
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Springer
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Discrete and Computational Geometry
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0179-5376