Equality Conditions for the Fractional Superadditive Volume Inequalities.
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| Title: | Equality Conditions for the Fractional Superadditive Volume Inequalities. |
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| 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] |
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| Database: | Engineering Source |
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| 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] |
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| ISSN: | 01795376 |
| DOI: | 10.1007/s00454-024-00672-8 |
