Sharp Affine LP Sobolev Inequalities

Lutwak, Erwin; Yang, Deane; Zhang, Gaoyong

doi:10.4310/jdg/1090425527

Cited by 302 publications

(242 citation statements)

References 28 publications

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“…for f ∈ W 1,p (R n ), 1 ≤ p < n, established by Lutwak, Yang and Zhang [33] for 1 < p < n and Zhang [45] for p = 1. Here W 1,p (R n ) is the usual Sobolev space defined as the set of functions f ∈ L p (R n ) with weak derivative ∇f ∈ L p (R n ).…”

Section: Introductionmentioning

confidence: 97%

Note on Affine Gagliardo–Nirenberg Inequalities

Zhai

2010

Potential Anal

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Abstract. This note proves sharp affine Gagliardo-Nirenberg inequalities which are stronger than all known sharp Euclidean Gagliardo-Nirenberg inequalities and imply the affine L p −Sobolev inequalities. The logarithmic version of affine L p −Sobolev inequalities is verified. Moreover, An alternative proof of the affine Moser-Trudinger and Morrey-Sobolev inequalities is given. The main tools are the equimeasurability of rearrangements and the strengthened version of the classical Pólys-Szegö principle.

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Section: Introductionmentioning

confidence: 97%

Note on Affine Gagliardo–Nirenberg Inequalities

Zhai

2010

Potential Anal

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“…During the past decade various elements of the L p Brunn-Minkowski theory have attracted increased attention (see e.g. [3], [4], [5], [8], [9] [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [22], [24], [25], [26], [27], [28], [29]). …”

mentioning

confidence: 99%

“…A solution to the L p Minkowski problem when the data is even was given in [12]. This solution turned out to be a critical ingredient in the recently established L p affine Sobolev inequality [18].…”

mentioning

confidence: 99%

On the Lp Minkowski Problem for Polytopes

Hug

Lutwak²,

Yang³

et al. 2005

Discrete Comput Geom

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Abstract. Two new approaches are presented to establish the existence of polytopal solutions to the discrete-data L p Minkowski problem for all p > 1.As observed by Schneider [23], the Brunn-Minkowski theory springs from joining the notion of ordinary volume in Euclidean d-space, R d , with that of Minkowski combinations of convex bodies. One of the cornerstones of the Brunn-Minkowski theory is the classical Minkowski problem. For polytopes the problem asks for the necessary and sufficient conditions on a set of unit vectors u 1 , . . . , u n ∈ S d−1 and a set of real numbers α 1 , . . . , α n > 0 that guarantee the existence of a polytope, P , in R d with n facets whose outer unit normals are u 1 , . . . , u n and such that the facet whose outer unit normal is u i has area (i.e.,

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“…The solutions to the Minkowski problem and the L p Minkowski problem connect with some important flows (see, e.g., [2,3,9,12,31]), and have important applications to Sobolev-type inequalities, see, e.g., Zhang [58], Lutwak, et al [43], Ciachi, et al [11], Haberl and Schuster [24][25][26], and Wang [57].…”

Section: Introductionmentioning

confidence: 99%