An extension of Berwald's inequality and its relation to Zhang's inequality

Alonso–Gutiérrez, David; Bernués, Julio; Merino, Bernardo González

doi:10.1016/j.jmaa.2020.123875

Cited by 12 publications

(5 citation statements)

References 12 publications

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“…In convex geometry, the Brunn-Minkowski theory (and its extensions) for convex bodies encompasses a large and growing range of fundamental results on the algebraic and geometric properties of convex bodies, and hence to study the parallel algebraic and geometric properties of log-concave functions is of great significance and in great demand. Recent years have witnessed that many results and notions in the Brunn-Minkowski theory (and its extensions) for convex bodies have found their functional analogues, including but not limited to the functional Blaschke-Santaló type inequality and its inverse [8,11,13,14,33,34,42], (John, Lutwak-Yang-Zhang, and Löwner) ellipsoids for log-concave functions [6,31,45], Rogers-Shephard type inequality and its reverse for log-concave functions [1,2,5,26], the affine surface areas for log-concave functions [9,20,21,22,23,44], the variation and Minkowski type problems related to log-concave functions [28,29,41,59], and (isoperimetric) inequalities related to log-concave functions [3,4,7,16,19,46,58]. Other contributions include e.g., [10,12,51] among others.…”

Section: Introduction and Overview Of The Main Resultsmentioning

confidence: 99%

Geometry of log-concave functions: the $L_p$ Asplund sum and the $L_{p}$ Minkowski problem

Fang¹,

Xing²,

Ye³

2020

Preprint

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The aim of this paper is to develop a basic framework of the L p theory for the geometry of log-concave functions, which can be viewed as a functional "lifting" of the L p Brunn-Minkowski theory for convex bodies. To fulfill this goal, by combining the L p Asplund sum of log-concave functions for all p > 1 and the total mass, we obtain a Prékopa-Leindler type inequality and propose a definition for the first variation of the total mass in the L p setting. Based on these, we further establish an L p Minkowski type inequality related to the first variation of the total mass and derive a variational formula which motivates the definition of our L p surface area measure for log-concave functions. Consequently, the L p Minkowski problem for log-concave functions, which aims to characterize the L p surface area measure for log-concave functions, is introduced. The existence of solutions to the L p Minkowski problem for log-concave functions is obtained for p > 1 under some mild conditions on the pre-given Borel measures.

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Section: Introduction and Overview Of The Main Resultsmentioning

confidence: 99%

Geometry of log-concave functions: the $L_p$ Asplund sum and the $L_{p}$ Minkowski problem

Fang¹,

Xing²,

Ye³

2020

Preprint

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“…A classical result due to Berwald [11] (see also [4,5] for other extensions and considerations), from which the Rogers-Shephard inequalities (1.2) and (1.9) can be derived, relates certain weighted power means of a concave function, as follows:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On Rogers–Shephard-type inequalities for the lattice point enumerator

Alonso–Gutiérrez

Lucas

Nicolás

2022

Commun. Contemp. Math.

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In this paper, we study various Rogers–Shephard-type inequalities for the lattice point enumerator [Formula: see text] on [Formula: see text]. In particular, for any non-empty convex bounded sets [Formula: see text], we show that [Formula: see text] and [Formula: see text] for [Formula: see text], [Formula: see text]. Additionally, a discrete counterpart to a classical result by Berwald for concave functions, from which other discrete Rogers–Shephard-type inequalities may be derived, is shown. Furthermore, we prove that these new discrete analogues for [Formula: see text] imply the corresponding results involving the Lebesgue measure.

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“…The following result can be seen as a degenerate version of Lemma 2.1 and its proof can be found in [2, Lemma 2.1]. In this case, we can characterise the equality cases.…”

Section: Preliminariesmentioning

confidence: 94%

“…Functional versions of the above for functions in

F ({double-struckR}^{n})

, the set of integrable log‐concave functions on

R^{n}

, was proved in [1, Lemma 3.3] for the range

γ > 0

, and was extended to the range

γ > - 1

in [2, Theorem 1.1]. Lemma Let

f \in F ({double-struckR}^{n})

and let

C

be the convex set

C = {false(x, t false) \in {double-struckR}^{n} \times false[0, \infty false) : f false(x false) ⩾ e^{- t} ∥ f {false∥}_{\infty}}

.…”

Section: Preliminariesmentioning

confidence: 99%

On affine invariant and local Loomis–Whitney type inequalities

Alonso–Gutiérrez

Bernués

Brazitikos

et al. 2020

J. London Math. Soc.

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We prove various extensions of the Loomis-Whitney inequality and its dual, where the subspaces on which the projections (or sections) are considered are either spanned by vectors w i of a not necessarily orthonormal basis of R n , or their orthogonal complements. In order to prove such inequalities we estimate the constant in the Brascamp-Lieb inequality in terms of the vectors w i. Restricted and functional versions of the inequality will also be considered.

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An extension of Berwald's inequality and its relation to Zhang's inequality

Cited by 12 publications

References 12 publications

Geometry of log-concave functions: the $L_p$ Asplund sum and the $L_{p}$ Minkowski problem

Geometry of log-concave functions: the $L_p$ Asplund sum and the $L_{p}$ Minkowski problem

On Rogers–Shephard-type inequalities for the lattice point enumerator

On affine invariant and local Loomis–Whitney type inequalities

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