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Equality Conditions for the Fractional Superadditive Volume Inequalities.

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Title: Equality Conditions for the Fractional Superadditive Volume Inequalities.
Authors: Meyer, Mark1 (AUTHOR) memeyer@shockers.wichita.edu
Source: Discrete & Computational Geometry. Jul2025, Vol. 74 Issue 1, p242-269. 28p.
Subjects: Lebesgue measure, Set functions, Generalization
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 A 1 , ⋯ , 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. [ABSTRACT FROM AUTHOR]
Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Equality Conditions for the Fractional Superadditive Volume Inequalities.
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  Data: 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 A 1 , ⋯ , 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. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s00454-024-00672-8
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              Text: Jul2025
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