Strengthened volume inequalities for $L_p$ zonoids of even isotropic measures

Böröczky, Károly J.; Fodor, Ferenc; Hug, Daniel

doi:10.1090/tran/7299

Cited by 3 publications

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References 53 publications

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“…Schneider [46] and Martinez-Maure [39] provide stability versions of the Alexandrov-Fenchel inequality if the bodies involved have C 2 + boundaries. For some additional recent related stability results, see [34,11,13,14].…”

Section: Introductionmentioning

confidence: 99%

Reverse Alexandrov–Fenchel inequalities for zonoids

Böröczky

Hug

2021

Commun. Contemp. Math.

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The Alexandrov–Fenchel inequality bounds from below the square of the mixed volume [Formula: see text] of convex bodies [Formula: see text] in [Formula: see text] by the product of the mixed volumes [Formula: see text] and [Formula: see text]. As a consequence, for integers [Formula: see text] with [Formula: see text] the product [Formula: see text] of suitable powers of the volumes [Formula: see text] of the convex bodies [Formula: see text], [Formula: see text], is a lower bound for the mixed volume [Formula: see text], where [Formula: see text] is the multiplicity with which [Formula: see text] appears in the mixed volume. It has been conjectured by Betke and Weil that there is a reverse inequality, that is, a sharp upper bound for the mixed volume [Formula: see text] in terms of the product of the intrinsic volumes [Formula: see text], for [Formula: see text]. The case where [Formula: see text], [Formula: see text], [Formula: see text] has recently been settled by the present authors (2020). The case where [Formula: see text], [Formula: see text], [Formula: see text] has been treated by Artstein-Avidan et al. under the assumption that [Formula: see text] is a zonoid and [Formula: see text] is the Euclidean unit ball. The case where [Formula: see text], [Formula: see text] is the unit ball and [Formula: see text] are zonoids has been considered by Hug and Schneider. Here, we substantially generalize these previous contributions, in cases where most of the bodies are zonoids, and thus we provide further evidence supporting the conjectured reverse Alexandrov–Fenchel inequality. The equality cases in all considered inequalities are characterized. More generally, stronger stability results are established as well.

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Section: Introductionmentioning

confidence: 99%

Reverse Alexandrov–Fenchel inequalities for zonoids

Böröczky

Hug

2021

Commun. Contemp. Math.

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Geometric and Functional Inequalities

Colesanti,

Hug

2023

Lecture Notes in Mathematics

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Reverse Alexandrov--Fenchel inequalities for zonoids

Böröczky¹,

Hug²

2021

Preprint

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The Alexandrov-Fenchel inequality bounds from below the square of the mixed volume V (K1, K2, K3, . . . , Kn) of convex bodies K1, . . . , Kn in R n by the product of the mixed volumes V (K1, K1, K3, . . . , Kn) and V (K2, K2, K3, . . . , Kn). As a consequence, for integers α1, . . . , αm ∈of suitable powers of the volumes Vn(Ki) of the convex bodies Ki, i = 1, . . . , m, is a lower bound for the mixed volume V (K1[α1], . . . , Km[αm]), where αi is the multiplicity with which Ki appears in the mixed volume. It has been conjectured by Ulrich Betke and Wolfgang Weil that there is a reverse inequality, that is, a sharp upper bound for the mixed volume V (K1[α1], . . . , Km[αm]) in terms of the product of the intrinsic volumes Vα i (Ki), for i = 1, . . . , m. The case where m = 2, α1 = 1, α2 = n − 1 has recently been settled by the present authors (2020). The case where m = 3, α1 = α2 = 1, α3 = n − 2 has been treated by Artstein-Avidan, Florentin, Ostrover (2014) under the assumption that K2 is a zonoid and K3 is the Euclidean unit ball. The case where α2 = • • • = αm = 1, K1 is the unit ball and K2, . . . , Km are zonoids has been considered by Hug, Schneider (2011). Here we substantially generalize these previous contributions, in cases where most of the bodies are zonoids, and thus we provide further evidence supporting the conjectured reverse Alexandrov-Fenchel inequality. The equality cases in all considered inequalities are characterized. More generally, stronger stability results are established as well.

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Strengthened volume inequalities for $L_p$ zonoids of even isotropic measures

Cited by 3 publications

References 53 publications

Reverse Alexandrov–Fenchel inequalities for zonoids

Reverse Alexandrov–Fenchel inequalities for zonoids

Geometric and Functional Inequalities

Reverse Alexandrov--Fenchel inequalities for zonoids

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