On the volume of the Minkowski sum of zonoids

Fradelizi, Matthieu; Madiman, Mokshay; Meyer, Mathieu; Zvavitch, Artem

doi:10.1016/j.jfa.2023.110247

Cited by 3 publications

(1 citation statement)

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“…• It is natural to ask what the analogue of the Lyusternik region looks like when, instead of allowing all compact sets, one restricts to convex sets. In this case, the question becomes clearly related to mixed volumes and their properties-indeed, supermodularity properties of mixed volumes are discussed in [18], some properties of the reverse kind (log-submodularity) that hold for special subclasses of convex sets are discussed in [18,17], and some extensions to measures beyond Lebesgue measure are discussed in [14]. We remark that studies of regions involving the set of possible mixed volumes of convex bodies have been undertaken in a series of works in convex geometry (see, e.g., [44,23]); however there does not appear to be a direct connection between our work and those results because our interest is focused on what can be said for general compact sets.…”

Section: Concluding Remarks and Open Questionsmentioning

confidence: 99%

Volumes of Subset Minkowski Sums and the Lyusternik Region

Barthe,

Madiman

2023

Discrete Comput Geom

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Section: Concluding Remarks and Open Questionsmentioning

confidence: 99%

Volumes of Subset Minkowski Sums and the Lyusternik Region

Barthe,

Madiman

2023

Discrete Comput Geom

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Sumset Estimates in Convex Geometry

Fradelizi,

Madiman,

Zvavitch

2024

International Mathematics Research Notices

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Sumset estimates, which provide bounds on the cardinality of sumsets of finite sets in a group, form an essential part of the toolkit of additive combinatorics. In recent years, probabilistic or entropic forms of many of these inequalities were introduced. We study analogues of these sumset estimates in the context of convex geometry and the Lebesgue measure on ${\mathbb{R}}^{n}$. First, we observe that, with respect to Minkowski summation, volume is supermodular to arbitrary order on the space of convex bodies. Second, we explore sharp constants in the convex geometry analogues of variants of the Plünnecke-Ruzsa inequalities. In the last section of the paper, we provide connections of these inequalities to the classical Rogers-Shephard inequality.

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