The logarithmic Minkowski problem

Böröczky, Károly; Lutwak, Erwin; Yang, Deane; Zhang, Gaoyong

doi:10.1090/S0894-0347-2012-00741-3

The logarithmic Minkowski problem

Authors: Károly J. Böröczky, Erwin Lutwak, Deane Yang and Gaoyong Zhang
Journal: J. Amer. Math. Soc. 26 (2013), 831-852
MSC (2010): Primary 52A40
DOI: https://doi.org/10.1090/S0894-0347-2012-00741-3
Published electronically: June 5, 2012
MathSciNet review: 3037788
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Abstract: 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.

References

A. Alexandroff, Existence and uniqueness of a convex surface with a given integral curvature, C. R. (Doklady) Acad. Sci. URSS (N.S.) 35 (1942), 131–134. MR 0007625
Ben Andrews, Gauss curvature flow: the fate of the rolling stones, Invent. Math. 138 (1999), no. 1, 151–161. MR 1714339, DOI https://doi.org/10.1007/s002220050344
Ben Andrews, Classification of limiting shapes for isotropic curve flows, J. Amer. Math. Soc. 16 (2003), no. 2, 443–459. MR 1949167, DOI https://doi.org/10.1090/S0894-0347-02-00415-0
Franck Barthe, Olivier Guédon, Shahar Mendelson, and Assaf Naor, A probabilistic approach to the geometry of the $l^n_p$-ball, Ann. Probab. 33 (2005), no. 2, 480–513. MR 2123199, DOI https://doi.org/10.1214/009117904000000874
Luis A. Caffarelli, Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation, Ann. of Math. (2) 131 (1990), no. 1, 135–150. MR 1038360, DOI https://doi.org/10.2307/1971510
S. Campi and P. Gronchi, The $L^p$-Busemann-Petty centroid inequality, Adv. Math. 167 (2002), no. 1, 128–141. MR 1901248, DOI https://doi.org/10.1006/aima.2001.2036
Wenxiong Chen, $L_p$ Minkowski problem with not necessarily positive data, Adv. Math. 201 (2006), no. 1, 77–89. MR 2204749, DOI https://doi.org/10.1016/j.aim.2004.11.007
Shiu Yuen Cheng and Shing Tung Yau, On the regularity of the solution of the $n$-dimensional Minkowski problem, Comm. Pure Appl. Math. 29 (1976), no. 5, 495–516. MR 423267, DOI https://doi.org/10.1002/cpa.3160290504
Kai-Seng Chou and Xu-Jia Wang, The $L_p$-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math. 205 (2006), no. 1, 33–83. MR 2254308, DOI https://doi.org/10.1016/j.aim.2005.07.004
Bennett Chow, Deforming convex hypersurfaces by the $n$th root of the Gaussian curvature, J. Differential Geom. 22 (1985), no. 1, 117–138. MR 826427
Andrea Cianchi, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Affine Moser-Trudinger and Morrey-Sobolev inequalities, Calc. Var. Partial Differential Equations 36 (2009), no. 3, 419–436. MR 2551138, DOI https://doi.org/10.1007/s00526-009-0235-4
Andrea Colesanti and Michele Fimiani, The Minkowski problem for torsional rigidity, Indiana Univ. Math. J. 59 (2010), no. 3, 1013–1039. MR 2779070, DOI https://doi.org/10.1512/iumj.2010.59.3937
William J. Firey, Shapes of worn stones, Mathematika 21 (1974), 1–11. MR 362045, DOI https://doi.org/10.1112/S0025579300005714
M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), no. 1, 69–96. MR 840401
Richard J. Gardner, Geometric tomography, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, New York, 2006. MR 2251886
R. J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 3, 355–405. MR 1898210, DOI https://doi.org/10.1090/S0273-0979-02-00941-2
R. J. Gardner and Gaoyong Zhang, Affine inequalities and radial mean bodies, Amer. J. Math. 120 (1998), no. 3, 505–528. MR 1623396
M. Gromov and V. D. Milman, Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces, Compositio Math. 62 (1987), no. 3, 263–282. MR 901393
Peter M. Gruber, Convex and discrete geometry, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 336, Springer, Berlin, 2007. MR 2335496
Bo Guan and Pengfei Guan, Convex hypersurfaces of prescribed curvatures, Ann. of Math. (2) 156 (2002), no. 2, 655–673. MR 1933079, DOI https://doi.org/10.2307/3597202

P. Guan and C.-S. Lin, On equation $\det (u_{i\!j}+\delta _{i\!j}u)=u^pf$ on $S^n$, preprint No. 2000-7, NCTS in Tsing-Hua University, 2000.

Pengfei Guan and Xi-Nan Ma, The Christoffel-Minkowski problem. I. Convexity of solutions of a Hessian equation, Invent. Math. 151 (2003), no. 3, 553–577. MR 1961338, DOI https://doi.org/10.1007/s00222-002-0259-2

Christoph Haberl, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, The even Orlicz Minkowski problem, Adv. Math. 224 (2010), no. 6, 2485–2510. MR 2652213, DOI https://doi.org/10.1016/j.aim.2010.02.006

Christoph Haberl and Franz E. Schuster, General $L_p$ affine isoperimetric inequalities, J. Differential Geom. 83 (2009), no. 1, 1–26. MR 2545028

Christoph Haberl and Franz E. Schuster, Asymmetric affine $L_p$ Sobolev inequalities, J. Funct. Anal. 257 (2009), no. 3, 641–658. MR 2530600, DOI https://doi.org/10.1016/j.jfa.2009.04.009

C. Haberl, F. Schuster, and J. Xiao, An asymmetric affine Pólya-Szegő principle, Math. Ann. 352 (2012), 517-542.

Binwu He, Gangsong Leng, and Kanghai Li, Projection problems for symmetric polytopes, Adv. Math. 207 (2006), no. 1, 73–90. MR 2264066, DOI https://doi.org/10.1016/j.aim.2005.11.006

Changqing Hu, Xi-Nan Ma, and Chunli Shen, On the Christoffel-Minkowski problem of Firey’s $p$-sum, Calc. Var. Partial Differential Equations 21 (2004), no. 2, 137–155. MR 2085300, DOI https://doi.org/10.1007/s00526-003-0250-9

Daniel Hug, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, On the $L_p$ Minkowski problem for polytopes, Discrete Comput. Geom. 33 (2005), no. 4, 699–715. MR 2132298, DOI https://doi.org/10.1007/s00454-004-1149-8

David Jerison, A Minkowski problem for electrostatic capacity, Acta Math. 176 (1996), no. 1, 1–47. MR 1395668, DOI https://doi.org/10.1007/BF02547334

Mei-Yue Jiang, Remarks on the 2-dimensional $L_p$-Minkowski problem, Adv. Nonlinear Stud. 10 (2010), no. 2, 297–313. MR 2656684, DOI https://doi.org/10.1515/ans-2010-0204

Fritz John, Polar correspondence with respect to a convex region, Duke Math. J. 3 (1937), no. 2, 355–369. MR 1545993, DOI https://doi.org/10.1215/S0012-7094-37-00327-2

Daniel A. Klain, The Minkowski problem for polytopes, Adv. Math. 185 (2004), no. 2, 270–288. MR 2060470, DOI https://doi.org/10.1016/j.aim.2003.07.001

Hans Lewy, On differential geometry in the large. I. Minkowski’s problem, Trans. Amer. Math. Soc. 43 (1938), no. 2, 258–270. MR 1501942, DOI https://doi.org/10.1090/S0002-9947-1938-1501942-3

Monika Ludwig, Ellipsoids and matrix-valued valuations, Duke Math. J. 119 (2003), no. 1, 159–188. MR 1991649, DOI https://doi.org/10.1215/S0012-7094-03-11915-8

Monika Ludwig, General affine surface areas, Adv. Math. 224 (2010), no. 6, 2346–2360. MR 2652209, DOI https://doi.org/10.1016/j.aim.2010.02.004

Monika Ludwig and Matthias Reitzner, A classification of ${\rm SL}(n)$ invariant valuations, Ann. of Math. (2) 172 (2010), no. 2, 1219–1267. MR 2680490, DOI https://doi.org/10.4007/annals.2010.172.1223

Erwin Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), no. 1, 131–150. MR 1231704

Erwin Lutwak and Vladimir Oliker, On the regularity of solutions to a generalization of the Minkowski problem, J. Differential Geom. 41 (1995), no. 1, 227–246. MR 1316557

Erwin Lutwak, Deane Yang, and Gaoyong Zhang, $L_p$ affine isoperimetric inequalities, J. Differential Geom. 56 (2000), no. 1, 111–132. MR 1863023

Erwin Lutwak, Deane Yang, and Gaoyong Zhang, A new ellipsoid associated with convex bodies, Duke Math. J. 104 (2000), no. 3, 375–390. MR 1781476, DOI https://doi.org/10.1215/S0012-7094-00-10432-2

Erwin Lutwak, Deane Yang, and Gaoyong Zhang, The Cramer-Rao inequality for star bodies, Duke Math. J. 112 (2002), no. 1, 59–81. MR 1890647, DOI https://doi.org/10.1215/S0012-9074-02-11212-5

Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Sharp affine $L_p$ Sobolev inequalities, J. Differential Geom. 62 (2002), no. 1, 17–38. MR 1987375

Erwin Lutwak, Deane Yang, and Gaoyong Zhang, On the $L_p$-Minkowski problem, Trans. Amer. Math. Soc. 356 (2004), no. 11, 4359–4370. MR 2067123, DOI https://doi.org/10.1090/S0002-9947-03-03403-2

Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Volume inequalities for subspaces of $L_p$, J. Differential Geom. 68 (2004), no. 1, 159–184. MR 2152912

Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Optimal Sobolev norms and the $L^p$ Minkowski problem, Int. Math. Res. Not. , posted on (2006), Art. ID 62987, 21. MR 2211138, DOI https://doi.org/10.1155/IMRN/2006/62987

Erwin Lutwak and Gaoyong Zhang, Blaschke-Santaló inequalities, J. Differential Geom. 47 (1997), no. 1, 1–16. MR 1601426

H. Minkowski, Allgemeine Lehrsätze über die convexen Polyeder, Nachr. Ges. Wiss. Göttingen, 198–219. Gesammelte Abhandlungen, Vol. II, Teubner, Leipzig, 1911, pp. 103–121.

Assaf Naor, The surface measure and cone measure on the sphere of $l_p^n$, Trans. Amer. Math. Soc. 359 (2007), no. 3, 1045–1079. MR 2262841, DOI https://doi.org/10.1090/S0002-9947-06-03939-0

Assaf Naor and Dan Romik, Projecting the surface measure of the sphere of $\scr l_p^n$, Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), no. 2, 241–261 (English, with English and French summaries). MR 1962135, DOI https://doi.org/10.1016/S0246-0203%2802%2900008-0

Louis Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337–394. MR 58265, DOI https://doi.org/10.1002/cpa.3160060303

G. Paouris, Concentration of mass on convex bodies, Geom. Funct. Anal. 16 (2006), no. 5, 1021–1049. MR 2276533, DOI https://doi.org/10.1007/s00039-006-0584-5

G. Paouris and E. Werner, Relative entropy of cone measures and $L_p$ centroid bodies, Proc. London Math. Soc. 104 (2012), 253–286.

Aleksey Vasil′yevich Pogorelov, The Minkowski multidimensional problem, V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto-London, 1978. Translated from the Russian by Vladimir Oliker; Introduction by Louis Nirenberg; Scripta Series in Mathematics. MR 0478079

Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521

Alina Stancu, The discrete planar $L_0$-Minkowski problem, Adv. Math. 167 (2002), no. 1, 160–174. MR 1901250, DOI https://doi.org/10.1006/aima.2001.2040

Alina Stancu, On the number of solutions to the discrete two-dimensional $L_0$-Minkowski problem, Adv. Math. 180 (2003), no. 1, 290–323. MR 2019226, DOI https://doi.org/10.1016/S0001-8708%2803%2900005-7

A. Stancu, Centro-affine invariants for smooth convex bodies, Int. Math. Res. Not. 2012, 2289–2320.

A. C. Thompson, Minkowski geometry, Encyclopedia of Mathematics and its Applications, vol. 63, Cambridge University Press, Cambridge, 1996. MR 1406315

Kaising Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math. 38 (1985), no. 6, 867–882. MR 812353, DOI https://doi.org/10.1002/cpa.3160380615

Vladimir Umanskiy, On solvability of two-dimensional $L_p$-Minkowski problem, Adv. Math. 180 (2003), no. 1, 176–186. MR 2019221, DOI https://doi.org/10.1016/S0001-8708%2802%2900101-9

Ge Xiong, Extremum problems for the cone volume functional of convex polytopes, Adv. Math. 225 (2010), no. 6, 3214–3228. MR 2729006, DOI https://doi.org/10.1016/j.aim.2010.05.016

Gaoyong Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999), no. 1, 183–202. MR 1776095