The even Orlicz Minkowski problem.(English)Zbl 1198.52003
For a convex body \(K\), let \(h_K\) and \(S_K\) denote its support function and its surface area measure, respectively. Let \(\varphi:(0, \infty)\rightarrow (0, \infty)\) be a fixed continuous function.
The aim of the paper is the study of the even \(L_{\varphi}\) Minkowski problem. Let \(\mu\) be an even finite Borel measure on the sphere which is not concentrated on a great subsphere. Does there exist an origin symmetric convex body \(K\) in \(\mathbb R^n\) such that
\[c\psi(h_K)dS_K=d\mu\; \]
for some positive number \(c\)?
The main results contained in the paper prove the existence for the even Orlicz Minkowski problem. As particular cases of these results, a solution to the even Minkowski problem is provided, as well as the solution for the even volume-normalized \(L_p\) Minkowski problem for all positive \(p\), \(0<p\neq n\), and the solution to the even volume-normalized \(L_p\) Minkowski problem for all positive \(p\). The paper presents a new approach to both, the classical Minkowski problem and the even \(L_p\) Minkowski problem for \(p>0\).
The aim of the paper is the study of the even \(L_{\varphi}\) Minkowski problem. Let \(\mu\) be an even finite Borel measure on the sphere which is not concentrated on a great subsphere. Does there exist an origin symmetric convex body \(K\) in \(\mathbb R^n\) such that
\[c\psi(h_K)dS_K=d\mu\; \]
for some positive number \(c\)?
The main results contained in the paper prove the existence for the even Orlicz Minkowski problem. As particular cases of these results, a solution to the even Minkowski problem is provided, as well as the solution for the even volume-normalized \(L_p\) Minkowski problem for all positive \(p\), \(0<p\neq n\), and the solution to the even volume-normalized \(L_p\) Minkowski problem for all positive \(p\). The paper presents a new approach to both, the classical Minkowski problem and the even \(L_p\) Minkowski problem for \(p>0\).
Reviewer: Eugenia Saorín Gómez (Magdeburg)
MSC:
| 52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
Keywords:
Minkowski problem;\(L_p\) Minkowski problem;Aleksandrov body;Orlicz norm;surface area measure;support function;Petty projection inequalityCite
Full Text:DOI
References:
| [1] | Campi, S.; Gronchi, P., The \(L^p\)-Busemann-Petty centroid inequality, Adv. Math., 167, 128-141 (2002), MR1901248, Zbl 1002.52005 ·Zbl 1002.52005 |
| [2] | Campi, S.; Gronchi, P., On the reverse \(L^p\)-Busemann-Petty centroid inequality, Mathematika, 49, 1-11 (2002), MR2059037, Zbl 1056.52005 ·Zbl 1056.52005 |
| [3] | Campi, S.; Gronchi, P., On volume product inequalities for convex sets, Proc. Amer. Math. Soc., 134, 2393-2402 (2006), MR2213713, Zbl 1095.52002 ·Zbl 1095.52002 |
| [4] | Campi, S.; Gronchi, P., Volume inequalities for \(L_p\)-zonotopes, Mathematika, 53, 71-80 (2006), (2007), MR2304053, Zbl 1117.52011 ·Zbl 1117.52011 |
| [5] | Chen, W., \(L_p\) Minkowski problem with not necessarily positive data, Adv. Math., 201, 77-89 (2006), MR2204749, Zbl 1102.34023 ·Zbl 1102.34023 |
| [6] | Cheng, S.-Y.; Yau, S.-T., On the regularity of the solutions of the \(n\)-dimensional Minkowski problem, Comm. Pure Appl. Math., 29, 495-516 (1976), MR423267, Zbl 0363.53030 ·Zbl 0363.53030 |
| [7] | Chou, K.-S.; Wang, X.-J., The \(L_p\)-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math., 205, 33-83 (2006), MR2254308, Zbl pre05054348 ·Zbl 1245.52001 |
| [8] | Cianchi, A.; Lutwak, E.; Yang, D.; Zhang, G., Affine Moser-Trudinger and Morrey-Sobolev inequalities, Calc. Var. Partial Differential Equations, 36, 419-436 (2009) ·Zbl 1202.26029 |
| [9] | Dafnis, N.; Paouris, G., Small ball probability estimates, \( \Psi_2\)-behavior and the hyperplane conjecture, J. Funct. Anal., 258, 1933-1964 (2010) ·Zbl 1189.52004 |
| [10] | Fleury, B.; Guédon, O.; Paouris, G. A., A stability result for mean width of \(L_p\)-centroid bodies, Adv. Math., 214, 865-877 (2007), MR2349721, Zbl 1132.52012 ·Zbl 1132.52012 |
| [11] | Gardner, R. J., The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.), 39, 355-405 (2002), MR1898210, Zbl 1019.26008 ·Zbl 1019.26008 |
| [12] | Gardner, R. J., Geometric Tomography, Encyclopedia Math. Appl., vol. 58 (2006), Cambridge University Press: Cambridge University Press Cambridge ·Zbl 1102.52002 |
| [13] | Giannopoulos, A.; Pajor, A.; Paouris, G., A note on subgaussian estimates for linear functionals on convex bodies, Proc. Amer. Math. Soc., 135, 2599-2606 (2007), MR2302581, Zbl 1120.52003 ·Zbl 1120.52003 |
| [14] | Gruber, P. M., Convex and Discrete Geometry, Grundlehren Math. Wiss., vol. 336 (2007), Springer: Springer Berlin, MR2335496, Zbl 1139.52001 ·Zbl 1139.52001 |
| [16] | Haberl, C., \(L_p\) intersection bodies, Adv. Math., 217, 2599-2624 (2008), MR2397461, Zbl 1140.52003 ·Zbl 1140.52003 |
| [17] | Haberl, C., Star body valued valuations, Indiana Univ. Math. J., 58, 2253-2276 (2009) ·Zbl 1183.52003 |
| [18] | Haberl, C.; Ludwig, M., A characterization of \(L_p\) intersection bodies, Int. Math. Res. Not. IMRN, 17, 29 (2006), Article ID 10548, Zbl 1115.52006 ·Zbl 1115.52006 |
| [19] | Haberl, C.; Schuster, F., General \(L_p\) affine isoperimetric inequalities, J. Differential Geom., 83, 1-26 (2009) ·Zbl 1185.52005 |
| [20] | Haberl, C.; Schuster, F., Asymmetric affine \(L_p\) Sobolev inequalities, J. Funct. Anal., 257, 641-658 (2009) ·Zbl 1180.46023 |
| [21] | Haberl, C.; Schuster, F.; Xiao, J., An asymmetric affine Pólya-Szegö principle ·Zbl 1241.26014 |
| [22] | Hu, C.; Ma, X.-N.; Shen, C., On the Christoffel-Minkowski problem of Firey’s \(p\)-sum, Calc. Var. Partial Differential Equations, 21, 137-155 (2004), MR2085300, Zbl pre02113880 ·Zbl 1161.35391 |
| [23] | Hug, D.; Lutwak, E.; Yang, D.; Zhang, G., On the \(L_p\) Minkowski problem for polytopes, Discrete Comput. Geom., 33, 699-715 (2005), MR2132298, Zbl 1078.52008 ·Zbl 1078.52008 |
| [25] | Klain, D., The Minkowski problem for polytopes, Adv. Math., 185, 270-288 (2004), MR2060470, Zbl 1053.52015 ·Zbl 1053.52015 |
| [26] | Ludwig, M., Projection bodies and valuations, Adv. Math., 172, 158-168 (2002), MR1942402, Zbl 1019.52003 ·Zbl 1019.52003 |
| [27] | Ludwig, M., Valuations on polytopes containing the origin in their interiors, Adv. Math., 170, 239-256 (2002), MR1932331, Zbl 1015.52012 ·Zbl 1015.52012 |
| [28] | Ludwig, M., Ellipsoids and matrix-valued valuations, Duke Math. J., 119, 159-188 (2003), MR1991649, Zbl 1033.52012 ·Zbl 1033.52012 |
| [29] | Ludwig, M., Minkowski valuations, Trans. Amer. Math. Soc., 357, 4191-4213 (2005), MR2159706, Zbl 1077.52005 ·Zbl 1077.52005 |
| [30] | Ludwig, M., Intersection bodies and valuations, Amer. J. Math., 128, 1409-1428 (2006), MR2275906, Zbl 1115.52007 ·Zbl 1115.52007 |
| [33] | Lutwak, E., The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom., 38, 131-150 (1993), MR1231704, Zbl 0788.52007 ·Zbl 0788.52007 |
| [34] | Lutwak, E., The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas, Adv. Math., 118, 244-294 (1996), MR1378681, Zbl 0853.52005 ·Zbl 0853.52005 |
| [35] | Lutwak, E.; Oliker, V., On the regularity of solutions to a generalization of the Minkowski problem, J. Differential Geom., 41, 227-246 (1995), MR1316557, Zbl 0867.52003 ·Zbl 0867.52003 |
| [36] | Lutwak, E.; Zhang, G., Blaschke-Santalo inequalities, J. Differential Geom., 47, 1-16 (1997), MR1601426, Zbl 0906.52003 ·Zbl 0906.52003 |
| [37] | Lutwak, E.; Yang, D.; Zhang, G., \(L_p\) affine isoperimetric inequalities, J. Differential Geom., 56, 111-132 (2000), MR1863023, Zbl 1034.52009 ·Zbl 1034.52009 |
| [38] | Lutwak, E.; Yang, D.; Zhang, G., A new ellipsoid associated with convex bodies, Duke Math. J., 104, 375-390 (2000), MR1781476, Zbl 0974.52008 ·Zbl 0974.52008 |
| [39] | Lutwak, E.; Yang, D.; Zhang, G., The Cramer-Rao inequality for star bodies, Duke Math. J., 112, 59-81 (2002), MR1890647, Zbl 1021.52008 ·Zbl 1021.52008 |
| [40] | Lutwak, E.; Yang, D.; Zhang, G., Sharp affine \(L_p\) Sobolev inequalities, J. Differential Geom., 62, 17-38 (2002), MR1987375, Zbl 1073.46027 ·Zbl 1073.46027 |
| [41] | Lutwak, E.; Yang, D.; Zhang, G., On the \(L_p\)-Minkowski problem, Trans. Amer. Math. Soc., 356, 11, 4359-4370 (2004), MR2067123, Zbl 1069.52010 ·Zbl 1069.52010 |
| [42] | Lutwak, E.; Yang, D.; Zhang, G., Volume inequalities for subspaces of \(L_p\), J. Differential Geom., 68, 159-184 (2004), MR2152912, Zbl 1119.52006 ·Zbl 1119.52006 |
| [43] | Lutwak, E.; Yang, D.; Zhang, G., \(L^p\) John ellipsoids, Proc. Lond. Math. Soc., 90, 497-520 (2005), MR2142136, Zbl 1074.52005 ·Zbl 1074.52005 |
| [44] | Lutwak, E.; Yang, D.; Zhang, G., Optimal Sobolev norms and the \(L^p\) Minkowski problem, Int. Math. Res. Not. IMRN, 62987, 1-21 (2006), MR2211138, Zbl 1110.46023 ·Zbl 1110.46023 |
| [45] | Lutwak, E.; Yang, D.; Zhang, G., Volume inequalities for isotropic measures, Amer. J. Math., 129, 1711-1723 (2007), MR2369894, Zbl 1134.52010 ·Zbl 1134.52010 |
| [46] | Lutwak, E.; Yang, D.; Zhang, G., Orlicz projection bodies, Adv. Math., 223, 220-242 (2010) ·Zbl 1437.52006 |
| [48] | Martinez-Maure, Y., Hedgehogs and zonoids, Adv. Math., 158, 1-17 (2001), MR1814896, Zbl 0977.52010 ·Zbl 0977.52010 |
| [49] | Meyer, M.; Werner, E., On the \(p\)-affine surface area, Adv. Math., 152, 288-313 (2000), MR1764106, Zbl 0964.52005 ·Zbl 0964.52005 |
| [50] | Molchanov, I., Convex and star-shaped sets associated with multivariate stable distributions. I. Moments and densities, J. Multivariate Anal., 100, 2195-2213 (2009) ·Zbl 1196.60029 |
| [51] | Oliker, V., Generalized convex bodies and generalized envelopes, (Contemp. Math., vol. 140 (1992)), 105-113, MR1197592, Zbl 0789.52007 ·Zbl 0789.52007 |
| [52] | Paouris, G., On the \(\psi_2\)-behaviour of linear functionals on isotropic convex bodies, Studia Math., 168, 285-299 (2005), MR2146128, Zbl 1078.52501 ·Zbl 1078.52501 |
| [53] | Paouris, G., Concentration of mass on convex bodies, Geom. Funct. Anal., 16, 1021-1049 (2006), MR2276533, Zbl 1114.52004 ·Zbl 1114.52004 |
| [54] | Paouris, G., Concentration of mass on isotropic convex bodies, C. R. Math. Acad. Sci. Paris, 342, 179-182 (2006) ·Zbl 1087.52002 |
| [56] | Pogorelov, A. V., The Minkowski Multidimensional Problem (1978), V.H. Winston: V.H. Winston Washington, DC, MR478079, Zbl 0387.53023 ·Zbl 0387.53023 |
| [57] | Rao, M. M.; Ren, Z. D., Theory of Orlicz Spaces, Monogr. Textbooks Pure Appl. Math., vol. 146 (1991), Marcel Dekker: Marcel Dekker New York, MR1113700, Zbl 0724.46032 ·Zbl 0724.46032 |
| [58] | Ryabogin, D.; Zvavitch, A., The Fourier transform and Firey projections of convex bodies, Indiana Univ. Math. J., 53, 667-682 (2004), MR2086696, Zbl 1062.52004 ·Zbl 1062.52004 |
| [59] | Schneider, R., Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia Math. Appl., vol. 44 (1993), Cambridge University Press: Cambridge University Press Cambridge, MR1216521, Zbl 0798.52001 ·Zbl 0798.52001 |
| [60] | Schütt, C.; Werner, E., Surface bodies and \(p\)-affine surface area, Adv. Math., 187, 98-145 (2004), MR2074173, Zbl 1089.52002 ·Zbl 1089.52002 |
| [61] | Stancu, A., The discrete planar \(L_0\)-Minkowski problem, Adv. Math., 167, 160-174 (2002), MR1901250, Zbl 1005.52002 ·Zbl 1005.52002 |
| [62] | Stancu, A., On the number of solutions to the discrete two-dimensional \(L_0\)-Minkowski problem, Adv. Math., 180, 290-323 (2003), MR2019226, Zbl 1054.52001 ·Zbl 1054.52001 |
| [63] | Stancu, A., The necessary condition for the discrete \(L_0\)-Minkowski problem in \(R^2\), J. Geom., 88, 162-168 (2008), MR2398486, Zbl 1132.52300 ·Zbl 1132.52300 |
| [64] | Thompson, A. C., Minkowski Geometry, Encyclopedia Math. Appl., vol. 63 (1996), Cambridge University Press: Cambridge University Press Cambridge, MR1406315, Zbl 0868.52001 ·Zbl 0868.52001 |
| [65] | Umanskiy, V., On solvability of two-dimensional \(L_p\)-Minkowski problem, Adv. Math., 180, 176-186 (2003), MR2019221, Zbl 1048.52001 ·Zbl 1048.52001 |
| [66] | Werner, E.; Ye, D., New \(L_p\) affine isoperimetric inequalities, Adv. Math., 218, 762-780 (2008), MR2414321, Zbl 1155.52002 ·Zbl 1155.52002 |
| [68] | Yaskin, V.; Yaskina, M., Centroid bodies and comparison of volumes, Indiana Univ. Math. J., 55, 1175-1194 (2006), MR2244603, Zbl 1102.52005 ·Zbl 1102.52005 |
| [69] | Zhang, G., The affine Sobolev inequality, J. Differential Geom., 53, 183-202 (1999), MR1776095, Zbl 1040.53089 ·Zbl 1040.53089 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
