The log-Brunn-Minkowski inequality in ℝ³

Yang, Yunlong; Zhang, Deyan

doi:10.1090/proc/14366

Cited by 16 publications

(4 citation statements)

References 29 publications

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“…These inequalities provide precisely "complementary" analogues of the log-Brunn-Minkowski and log-Minkowski inequalities for convex bodies conjectured by Böröczky, Lutwak, Yang, and Zhang in [5]. Note that the log-Brunn-Minkowski and log-Minkowski inequalities for convex bodies are still quite open in general and have received a lot of attention, see e.g., [4,32,39,46,50]. The significance of our log-Minkowski inequality for C-coconvex sets (i.e., (7.58)) can also be seen from the fact that this inequality gives a positive answer to an open problem raised by Schneider in [41].…”

Section: Introduction and Overview Of The Main Resultsmentioning

confidence: 93%

On the $L_p$ Brunn-Minkowski theory and the $L_p$ Minkowski problem for $C$-coconvex sets

Yang,

Ye,

Zhu

2022

Preprint

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Let C be a pointed closed convex cone in R n with vertex at the origin o and having nonempty interior. The set A ⊂ C is C-coconvex if the volume of A is finite and A • = C \ A is a closed convex set. For 0 < p < 1, the p-co-sum of C-coconvex sets is introduced, and the corresponding L p Brunn-Minkowski inequality for C-coconvex sets is established. We also define the L p surface area measures, for 0 = p ∈ R, of certain C-coconvex sets, which are critical in deriving a variational formula of the volume of the Wulff shape associated with a family of functions obtained from the p-co-sum. This motivates the L p Minkowski problem aiming to characterize the L p surface area measures of C-coconvex sets. The existence of solutions to the L p Minkowski problem for all 0 = p ∈ R is established. The L p Minkowski inequality for 0 < p < 1 is proved and is used to obtain the uniqueness of the solutions to the L p Minkowski problem for 0 < p < 1.For p = 0, we introduce (1 − τ ) A 1 ⊕ 0 τ A 2 , the log-co-sum of two C-coconvex sets A 1 and A 2 with respect to τ ∈ (0, 1), and prove the log-Brunn-Minkowski inequality of C-coconvex sets. The log-Minkowski inequality is also obtained and is applied to prove the uniqueness of the solutions to the log-Minkowski problem that characterizes the cone-volume measures of C-coconvex sets. Our result solves an open problem raised by Schneider in [

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Section: Introduction and Overview Of The Main Resultsmentioning

confidence: 93%

On the $L_p$ Brunn-Minkowski theory and the $L_p$ Minkowski problem for $C$-coconvex sets

Yang,

Ye,

Zhu

2022

Preprint

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“…A natural question is whether the Green-Osher inequality holds without symmetric condition. Similar question is asked by the log-Brunn-Minkowski inequality (see Böröczky-Lutwak-Yang-Zhang [1], Xi-Leng [11] and Yang-Zhang [13]). Xi and Leng [11] gave the definition of dilation position for the first time to prove the log-Brunn-Minkowski inequality and solve the planar Dar's conjecture.…”

Section: Introductionmentioning

confidence: 90%

Nonsymmetric extension of the Green–Osher inequality

Yang

2019

Geom Dedicata

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“…See the excellent survey article of Gardner [2] and the book of Schneider [3], which contains a comprehensive account of different aspects and consequences of Brunn-Minkowski inequality. More recent papers about Brunn-Minkowski-type inequalities include [4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

An Expository Lecture of María Jesús Chasco on Some Applications of Fubini’s Theorem

Castejón

Chasco

Corbacho

et al. 2021

Axioms

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The usefulness of Fubini’s theorem as a measurement instrument is clearly understood from its multiple applications in Analysis, Convex Geometry, Statistics or Number Theory. This article is an expository paper based on a master class given by the second author at the University of Vigo and is devoted to presenting some Applications of Fubini’s theorem. In the first part, we present Brunn–Minkowski’s and Isoperimetric inequalities. The second part is devoted to the estimations of volumes of sections of balls in Rn.

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The log-Brunn-Minkowski inequality in ℝ³

Cited by 16 publications

References 29 publications

On the $L_p$ Brunn-Minkowski theory and the $L_p$ Minkowski problem for $C$-coconvex sets

On the $L_p$ Brunn-Minkowski theory and the $L_p$ Minkowski problem for $C$-coconvex sets

Nonsymmetric extension of the Green–Osher inequality

An Expository Lecture of María Jesús Chasco on Some Applications of Fubini’s Theorem

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