On the log-Minkowski inequality for simplices and parallelepipeds

Henk, Martin; Pollehn, Hannes

doi:10.1007/s10474-018-0822-y

Cited by 13 publications

(7 citation statements)

References 45 publications

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“…So far, the Log-Minkowski inequality has been verified for all planar convex bodies in [3], for the class of unconditional bodies in [11] and for several other families of convex bodies -some of which are not centrally-symmetric -in [13] and [6]. We remark that the Log-Minkowski inequality does not hold for arbitrary non-symmetric convex bodies, as pointed out in [3].…”

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confidence: 73%

“…As observed in [6], an affirmative answer to Problem 1.2 would imply the Log-Minkowski inequality for a centered convex body and a centered parallelotope. An affirmative answer to this problem was confirmed in [6,Proposition 1.5] for n ≤ 4.…”

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confidence: 74%

“…The following quantitative isobarycentric problem was posed by Martin Henk, using a different normalization, and appears implicitly in [6]. It is this problem that sparked the research presented here.…”

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confidence: 99%

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Isobarycentric Inequalities

Gilboa,

Haim-Kislev,

Slomka

2022

Preprint

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We prove the following isoperimetric type inequality: Given a finite absolutely continuous Borel measure on R n , halfspaces have maximal measure among all subsets with prescribed barycenter. As a consequence, we make progress towards a solution to a problem of Henk and Pollehn, which is equivalent to a Log-Minkowski inequality for a parallelotope and a centered convex body. Our probabilistic approach to the problem also gives rise to several inequalities and conjectures concerning the truncated mean of certain log-concave random variables.

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confidence: 73%

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confidence: 74%

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Isobarycentric Inequalities

Gilboa,

Haim-Kislev,

Slomka

2022

Preprint

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“…Recently, the log-Minkowski inequality and the log-Aleksandrov-Fenchel inequality and its dual form have attracted extensive attention and research: see references [1,3,4,6,7,8,12,13,14,18,17,19,20,21,22,23]. In this paper, we generalize the log-Minkowski inequality (1.1) and the log-Aleksandrov-Fenchel inequality (1.2) to the mixed affine quermassintegrals.…”

Section: Introductionmentioning

confidence: 94%

The affine log-Aleksandrov–Fenchel inequality

Zhao¹

2023

Revista De La Unión Matemática Argentina

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We establish a new affine log-Aleksandrov-Fenchel inequality for mixed affine quermassintegrals by introducing new concepts of affine and multiple affine measures, and using the newly established Aleksandrov-Fenchel inequality for multiple mixed affine quermassintegrals. Our new inequality yields as special cases the classical Aleksandrov-Fenchel inequality and the Lp-affine log-Aleksandrov-Fenchel inequality. The affine log-Minkowski and log-Aleksandrov-Fenchel inequalities are also derived.

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“…The log-Minkowski inequality belongs to log-Minkowski theory. For more research on log-Minkowski theory, we may refer to [13][14][15][16][17][18][19][20][21][22].…”

Section: Theorem 1b (The Log-minkowski Inequality For Mixed Quermassmentioning

confidence: 99%

Log-Minkowski inequalities for the L p $L_{p}$ -mixed quermassintegrals

Wang

2019

J Inequal Appl

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Böröczky et al. proposed the log-Minkowski problem and established the plane log-Minkowski inequality for origin-symmetric convex bodies. Recently, Stancu proved the log-Minkowski inequality for mixed volumes; Wang, Xu, and Zhou gave the L p version of Stancu's results. In this paper, we define the L p-mixed quermassintegrals probability measure and obtain the log-Minkowski inequality for the L p-mixed quermassintegrals. As its application, we establish the L p-mixed affine isoperimetric inequality. In addition, we also consider the dual log-Minkowski inequalities for the L p-dual mixed quermassintegrals.

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On the log-Minkowski inequality for simplices and parallelepipeds

Cited by 13 publications

References 45 publications

Isobarycentric Inequalities

Isobarycentric Inequalities

The affine log-Aleksandrov–Fenchel inequality

Log-Minkowski inequalities for the L p $L_{p}$ -mixed quermassintegrals

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