Some remarks on Petty projection of log-concave functions

Silva, Leticia Alves da; Merino, Bernardo González; Villa, Rafael Duarte

doi:10.48550/arxiv.1906.08183

Cited by 3 publications

(6 citation statements)

References 13 publications

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“…By the above observation, f 0 is a weak solution of (13) in W 1,p 0 (Ω) when we take ρ as the constant function ρ(x) = λ A 1,p (Ω). Then, by Proposition 5, we deduce that (13), by Proposition 4, we conclude that f 0 is a bounded function in C 1,α (Ω) for arbitrary Ω and in C 1,α (Ω) if Ω has boundary of C 2,α class.…”

Section: Affine Invariance Properties Here We Deal With the Invarianc...mentioning

confidence: 79%

“…Assume p = 1 and f is the characteristic function of a convex body K, then L 1,f is, up to a constant depending on n and p, the polar projection body Π • K (see [56,Definition 10.77]) and inequality (4) becomes the Petty projection inequality, an affine-invariant version of the classical isoperimetric inequality. The set L p,f appears in the literature, sometimes with the notation Π • p f (see for example [1,13]) since it is a functional version of the polar projection operator. For a given convex body K ⊂ R n there are many bodies associated to it.…”

Section: Preliminaries On Convex Geometrymentioning

confidence: 99%

“…Proof. Using the strategy of the previous proof which consists in introducing the quasilinear elliptic operator L = L p,f0 on W 1,p 0 (Ω) for fixed f 0 , we rewrite (13) as…”

Section: Affine Invariance Properties Here We Deal With the Invarianc...mentioning

confidence: 99%

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From affine Poincaré inequalities to affine spectral inequalities

Haddad¹,

Jimenez²,

Montenegro³

2020

Preprint

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Given a bounded open subset Ω of R n , we establish the weak closure of the affine ball B A p (Ω) = {f ∈ W 1,p 0 (Ω) : Epf ≤ 1} with respect to the affine functional Epf introduced by Lutwak, Yang and Zhang in [43] as well as its compactness in L p (Ω) for any p ≥ 1. This part relies strongly on the celebrated Blaschke-Santaló inequality. As counterpart, we develop the basic theory of p-Rayleigh quotients in bounded domains, in the affine case, for p ≥ 1. More specifically, we establish p-affine versions of the Poincaré inequality and some of their consequences. We introduce the affine invariant p-Laplace operator ∆ A p f defining the Euler-Lagrange equation of the minimization problem of the p-affine Rayleigh quotient. We also study its first eigenvalue λ A 1,p (Ω) which satisfies the corresponding affine Faber-Krahn inequality, this is that λ A 1,p (Ω) is minimized (among sets of equal volume) only when Ω is an ellipsoid. This point depends fundamentally on PDEs regularity analysis aimed at the operator ∆ A p f . We also present some comparisons between affine and classical eigenvalues and characterize the cases of equality for p ≥ 1. All affine inequalities obtained are stronger and directly imply the classical ones.

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Section: Affine Invariance Properties Here We Deal With the Invarianc...mentioning

confidence: 79%

Section: Preliminaries On Convex Geometrymentioning

confidence: 99%

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From affine Poincaré inequalities to affine spectral inequalities

Haddad¹,

Jimenez²,

Montenegro³

2020

Preprint

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“…As mentioned in [14,Corollary 3.1] that the first inequality in Theorem 2.2 includes a geometric inequality, which involves the surface area and the volume of the polar body of Petty projection body. The second inequality in Theorem 2.2 implies the Petty projection inequality.…”

Section: Petty Projection Inequality For Log-concave Functionsmentioning

confidence: 99%

Notes on the functional LYZ ellipsoid

Fang,

Zhou

2019

Preprint

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“…The Minkowski functional of a convex body is a norm on R n if K has non-empty interior and K is symmetric. More recently, the polar projection LYZ bodies appeared in [47] under the name Petty Projection bodies, where Π f was denoted as 2Π b f . Furthermore, in [47], they extended Petty's projection inequality to the LYZ bodies of log-concave functions, where they proved the following:…”

Section: Introductionmentioning

confidence: 99%

General Measure Extensions of Projection Bodies

Langharst,

Roysdon,

Zvavitch

2021

Preprint

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The inequalities of Petty and Zhang are affine isoperimetric-type inequalities providing sharp bounds for Vol n−1 n (K)Vol n (Π • K), where ΠK is a projection body of a convex body K. In this paper, we present a number of generalizations of Zhang's inequality to the setting of arbitrary measures. In addition, we introduce extensions of the projection body operator Π to the setting of arbitrary measures and functions, while providing associated inequalities for this operator; in particular, Zhang-type inequalities. Throughout, we apply shown results to the standard Gaussian measure on R n .

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Some remarks on Petty projection of log-concave functions

Abstract: In this note we study the Petty projection of a log-concave function, which has been recently introduced in [9]. Moreover, we present some new inequalities involving this new notion, partly complementing and correcting some results from [9].

Cited by 3 publications

References 13 publications

From affine Poincaré inequalities to affine spectral inequalities

From affine Poincaré inequalities to affine spectral inequalities

Notes on the functional LYZ ellipsoid

General Measure Extensions of Projection Bodies

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