Optimal Sobolev norms and the Lp Minkowski problem

By Erwin Lutwak, Deane Yang, Gaoyong Zhang. 2006

174 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/02b7733bbc1cfd925f0c7494dbacc9150e5d8922

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

The $L_p$-Aleksandrov problem for $L_p$-integral curvature (2018, methodology): “In [11, 21, 37, 39, 54] affine Sobolev inequalities were obtained by using the solution of the Minkowski problem and the Lp-Minkowski problem (together with Lpaffine isoperimetric inequalities from [36, 20]).”
The dual Minkowski problem for symmetric convex bodies (2017, background): “ Lp-Minkowski problem, see B¨oro¨czky-Trinh [11] and Chen-Li-Zhu [12,13]. The Lp Minkowski problem also plays a key role in establishing aﬃne Sobolev inequalities (see, for example, Lutwak-Yang-Zhang [40,41], Cianchi-Lutwak-Yang-Zhang [14], and Haberl & Schuster [26]). Very recently, Huang-Lutwa”
On the uniqueness of Lp-Minkowski problems: The constant p-curvature case in R3 (2015, background): “Lutwak [28, 29] to the Brunn-Minkowski-Firey theory. Many interesting applications and inequalities have been correspondingly established following this pioneering development in convex geometry, see [5, 9, 32, 33, 34, 44] for example. Among many excellent references, we refer the reader to the newl”
The even Orlicz-Minkowski problem (2010, unknown): “[1–5, 7–10, 12, 13, 15–22, 25–39, 41–44, 47, 49, 50, 52–55, 58, 60–63, 65–68]).”
Embedding Sn into Rn+1 with given integral Gauss curvature and optimal mass transport on Sn (2007, methodology): “It is worthwhile noting that the variational approach used in this paper is quite general and is expected to be useful for studying other nonlinear geometric problems, in particular, the generalized Minkowski problem considered in [21], [22], [23].”
Sharp affine Sobolev type inequalities via the $\Lp$ Busemann-Petty centroid inequality (2015, unknown): “arp Sobolev inequalities obtained by means of a mass transportation method. The connection between the Lp extension of the Minkowski problem and sharp aﬃne Sobolev inequalities was already studied in [31]. Our approach also allows us to prove Theorem 1.4 with the same two ingredients mentioned above”
The logarithmic Minkowski problem (2012, unknown): “Connections of the L p -Minkowski problem with optimal Sobolev norms was shown in [47].”
New approach to the affine P\'olya-Szeg\"o principle and the stability version of the affine Sobolev inequality (2015, methodology): “s section some background material from the Lp Brunn-Minkowski theory of convex bodies. This theory has its origins in the work of Firey [27] and was further developed by Lutwak and many authors (see [9, 31, 32, 35, 42, 43, 44, 46, 47, 62]). We also list some basic facts from real analysis which we ”
Centro-Affine Invariants for Smooth Convex Bodies (2010, background): “The new view brought a renewed focus on the class of convex bodies (compact convex sets) containing the origin in their interior on which many affine invariants and many affineinvariant inequalities were derived, [6], [7], [12]–[15], [23], [31]–[35], [54]– [56].”
Affine Moser–Trudinger and Morrey–Sobolev inequalities (2009, background): “, [12], [14], [27], [31], [32], [33] – [41], [44], [48] – [50]).”