The logarithmic Minkowski problem

By K. Boroczky, Erwin Lutwak, Deane Yang, Gaoyong Zhang. 2012

440 citations recorded by the source corpus.

In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of a finite dimensional Banach space.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ac16355f95bec24902d4a92bfe48be5d80375b09

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

New Orlicz affine isoperimetric inequalities (2014, background): “Important contributions include the study of Logarithmic Minkowski and even Orlicz Minkowski problems [7, 15]; the logBrunn-Minkowski-inequality [5]; stronger versions of the Orlicz-Petty projection inequality [6]; Orlicz Busemann-Petty centroid inequality [10, 20, 53] among others.”
Cone-volume measure of general centered convex bodies (2016, neutral): “In the recent paper [11], K.”
More on logarithmic sums of convex bodies (2014, background): “In [5], a necessary and sufficient condition was discovered (the planar case was treated by Stancu [42] [43]; see also [6] for other applications of the cone-volume measure and [8] for a possible functional generalization of the classical Minkowski problem).”
The Gauss Image Problem (2020, methodology): “The techniques developed in this work in order to obtain these critical estimates are new and different from those developed in [16, 27].”
Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems (2016, positive): “If the measure µ satisﬁes the k -subspace mass inequality, then there exists an origin-symmetric convex body K in R n such that The case of k = n was proved in [14].”
On the L dual Minkowski problem (2018, methodology): “When the given data is even, the existence of solutions to the logarithmic Minkowski problem was completely solved in Böröczky, Lutwak, Yang, & Zhang [10].”
The dual Orlicz–Brunn–Minkowski theory ☆ (2014, background): “) The new theory has already attracted considerable interest; see, for example, [2, 3, 4, 5, 21, 22, 25, 26, 39, 40, 41, 46].”
Moment Measures (2013, methodology): “Zhang [6].”
On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalities (2015, methodology): “In [31] the proof is done by linking the problem to the new logBrunn-Minkowski inequality of Böröczky, Lutwak, Yang and Zhang; see [6], [7], [30] and [31].”
The discrete logarithmic Minkowski problem (2014, positive): “[59]). When p 6= 1, the Lp Minkowski problem and PDE (1.1) have been studied by, e.g., Lutwak [43], Lutwak and Oliker [44], Guan and Lin [21], Chou and Wang [14], Hug, et al. [33], B¨oro¨czky, et al. [6]. Additional references regarding the Lp Minkowski problem and Minkowski-type problems can be fou”
Remarks on the conjectured log-Brunn–Minkowski inequality (2013, background): “This problem has recently been solved by Böröczky, Lutwak, Yang and Zhang [3] in the case of even measures.”
The Lp Minkowski problem for polytopes for 0 < p < 1 (2014, neutral): “ [21]. Necessary and suﬃcient conditions for the existence of solutions to the even L0 Minkowski problem (also called the logarithmic Minkowski problem) was recently established by B¨oro¨czky, et al. [5]. Without the assumption that the measure is even, existence of solution of the PDE (1.1) for the”
A flow method for the dual Orlicz–Minkowski problem (2020, background): “[5, 7, 8, 15, 17, 28, 32, 36, 39, 40, 41, 47, 48, 49, 50, 52, 54, 67, 68] and Schneider’s book [59], and corresponding references therein.”