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Uniqueness and Continuity of the Solution to \(L_p\) Dual Minkowski Problem

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Abstract

Lutwak et al. (Adv Math 329:85–132, 2018) introduced the \(L_p\) dual curvature measure that unifies several other geometric measures in dual Brunn–Minkowski theory and Brunn–Minkowski theory. Motivated by works in Lutwak et al. (Adv Math 329:85–132, 2018), we consider the uniqueness and continuity of the solution to the \(L_p\) dual Minkowski problem. To extend the important work (Theorem A) of LYZ to the case for general convex bodies, we establish some new Minkowski-type inequalities which are closely related to the optimization problem associated with the \(L_p\) dual Minkowski problem. When \(q< p\), the uniqueness of the solution to the \(L_p\) dual Minkowski problem for general convex bodies is obtained. Moreover, we obtain the continuity of the solution to the \(L_p\) dual Minkowski problem for convex bodies.

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References

  1. Aleksandrov, A.D.: On the theory of mixed volumes. III. Extension of two theorems of Minkowski on convex polyhedra to arbitrary convex bodies. Mat. Sb.(NS) 3, 27–46 (1938)

    Google Scholar 

  2. Aleksandrov, A.D.: Existence and uniqueness of a convex surface with a given integral curvature. C. R. (Doklady) Acad. Sci. URSS (N.S.) 35, 131–134 (1942)

    MathSciNet  Google Scholar 

  3. Barthe, F., Guédon, O., Mendelson, S., Naor, A.: A probabilistic approach to the geometry of the \(l^n_p\)-ball. Ann. Probab. 33, 480–513 (2005)

    Article  MathSciNet  Google Scholar 

  4. Böröczky, K.J., Fodor, F.: On the \(L_p\) dual Minkowski problem for \(p>1\) and \(q>0\). J. Differ. Equ. 266, 7980–8033 (2019)

    Article  ADS  Google Scholar 

  5. Böröczky, K.J., Hegedűs, P., Zhu, G.: On the discrete logarithmic Minkowski problem. Int. Math. Res. Not. 6, 1807–1838 (2016)

    Article  MathSciNet  Google Scholar 

  6. Böröczky, K.J., Henk, M.: Cone-volume measure of general centered convex bodies. Adv. Math. 286, 703–721 (2016)

    Article  MathSciNet  Google Scholar 

  7. Böröczky, K.J., Henk, M., Pollehn, H.: Subspace concentration of dual curvature measures of symmetric convex bodies. J. Differ. Geom. 109, 411–429 (2018)

    Article  MathSciNet  Google Scholar 

  8. Böröczky, K.J., Lutwak, E., Yang, D., Zhang, G.: The log-Brunn–Minkowski inequality. Adv. Math. 231, 197–1997 (2012)

    Article  MathSciNet  Google Scholar 

  9. Böröczky, K.J., Lutwak, E., Yang, D., Zhang, G.: The logarithmic Minkowski problem. J. Am. Math. Soc. 26, 831–852 (2013)

    Article  MathSciNet  Google Scholar 

  10. Chen, C., Huang, Y., Zhao, Y.: Smooth solutions to the \(L_p\) dual Minkowski problem. Math. Ann. 373, 953–976 (2019)

    Article  MathSciNet  Google Scholar 

  11. Chen, S., Li, Q., Zhu, G.: On the \(L_p\) Monge-Ampère equation. J. Differ. Equ. 263, 4997–5011 (2017)

    Article  ADS  Google Scholar 

  12. Chen, W.: \(L_p\) Minkowski problem with not necessarily positive data. Adv. Math. 201, 77–89 (2006)

    Article  MathSciNet  Google Scholar 

  13. Cianchi, A., Lutwak, E., Yang, D., Zhang, G.: Affine Moser–Trudinger and Morrey–Sobolev inequalities. Calc. Var. Partial Differ. Equ. 36, 419–436 (2009)

    Article  MathSciNet  Google Scholar 

  14. Chou, K.S., Wang, X.J.: The \(L_p\)-Minkowski problem and the Minkowski problem in centroaffine geometry. Adv. Math. 205, 33–83 (2006)

    Article  MathSciNet  Google Scholar 

  15. Colesanti, A., Fragalà, I.: The first variation of the total mass of log-concave functions and related inequalities. Adv. Math. 244, 708–749 (2013)

    Article  MathSciNet  Google Scholar 

  16. Cordero-Erausquin, D., Klartag, B.: Moment measure. J. Funct. Anal. 268, 3834–3866 (2015)

    Article  MathSciNet  Google Scholar 

  17. Fang, N., Zhou, J.: LYZ ellipsoid and Petty projection body for log-concave functions. Adv. Math. 340, 914–959 (2018)

    Article  MathSciNet  Google Scholar 

  18. Fenchel, W., Jessen, B.: Mengenfunktionen und konvexe Körper, Danske Vid. Selskab. Mat.-fys. Medd. 16, 1–31 (1938)

    Google Scholar 

  19. Gardner, R.J., Hug, D., Weil, W., Xing, S., Ye, D.: General volumes in the Orlicz–Brunn–Minkowski theory and a related Minkowski problem I. Calc. Var. Partial Differ. Equ. 58(1), 12 (2019)

    Article  MathSciNet  Google Scholar 

  20. Gardner, R.J., Hug, D., Xing, S., Ye, D.: General volumes in the Orlicz–Brunn–Minkowski theory and a related Minkowski problem II. Calc. Var. Partial Differ. Equ. 59(1), 15 (2020)

    Article  MathSciNet  Google Scholar 

  21. Haberl, C., Lutwak, E., Yang, D., Zhang, G.: The even Orlicz Minkowski problem. Adv. Math. 224, 2485–2510 (2010)

    Article  MathSciNet  Google Scholar 

  22. Haberl, C., Schuster, F.E.: General \(L_p\) affine isoperimetric inequalities. J. Differ. Geom. 83, 1–26 (2009)

    Article  Google Scholar 

  23. Haberl, C., Schuster, F.E.: Asymmetric affine \(L_p\) Sobolev inequalities. J. Funct. Anal. 257, 641–658 (2009)

    Article  MathSciNet  Google Scholar 

  24. Haberl, C., Schuster, F.E., Xiao, J.: An asymmetric affine Pólya-Szegö principle. Math. Ann. 352, 517–542 (2012)

    Article  MathSciNet  Google Scholar 

  25. Huang, Q., He, B.: On the Orlicz Minkowski problem for polytopes. Discrete Comput. Geom. 48, 281–297 (2012)

    Article  MathSciNet  Google Scholar 

  26. Huang, Y., Liu, J., Xu, L.: On the uniqueness of \(L_p\)-Minkowski problems: the constant \(p\)-curvature case in \({\mathbb{R} }^3\). Adv. Math. 281, 906–927 (2015)

    Article  MathSciNet  Google Scholar 

  27. Huang, Y., Lutwak, E., Yang, D., Zhang, G.: Geometric measures in the dual Brunn–Minkowki theory and their associated Minkowski problems. Acta Math. 216, 325–388 (2016)

    Article  MathSciNet  Google Scholar 

  28. Huang, Y., Lutwak, E., Yang, D., Zhang, G.: The \(L_p\) Aleksandrov problem for \(L_p\) integral curvature. J. Differ. Geom. 110, 1–29 (2018)

    Article  ADS  Google Scholar 

  29. Huang, Y., Zhao, Y.: On the \(L_p\) dual Minkowski problem. Adv. Math. 332, 57–84 (2018)

    Article  MathSciNet  Google Scholar 

  30. Hug, D., Lutwak, E., Yang, D., Zhang, G.: On the \(L_p\) Minkowski problem for polytopes. Discrete Comput. Geom. 33, 699–715 (2005)

    Article  MathSciNet  Google Scholar 

  31. Jian, H., Lu, J., Zhu, G.: Mirror symmetric solutions to the centro-affine Minkowski problem. Calc. Var. Partial Differ. Equ. 55(2), 41 (2016)

    Article  MathSciNet  Google Scholar 

  32. Li, Q.-R., Liu, J., Lu, J.: Nonuniqueness of solutions to the \(L_p\) dual Minkowski problem. Int. Math. Res. Not. IMRN 2022, 9114–9150 (2022)

    Article  Google Scholar 

  33. Lv, S.: A functional Busemann intersection inequality. J. Geom. Anal. 31, 6274–6291 (2021)

    Article  MathSciNet  Google Scholar 

  34. Lu, J., Wang, X.J.: Rotationally symmetric solutions to the \(L_p\)-Minkowski problem. J. Differ. Equ. 254, 983–1005 (2013)

    Article  ADS  Google Scholar 

  35. Ludwig, M., Xiao, J., Zhang, G.: Sharp convex Lorentz–Sobolev inequalities. Math. Ann. 350, 169–197 (2011)

    Article  MathSciNet  Google Scholar 

  36. Lutwak, E.: Dual mixed volumes. Pacific J. Math. 58, 531–538 (1975)

    Article  MathSciNet  Google Scholar 

  37. Lutwak, E.: The Brunn–Minkowski–Firey theory. I. Mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131–150 (1993)

    Article  MathSciNet  Google Scholar 

  38. Lutwak, E., Oliker, V.: On the regularity of solutions to a generalization of the Minkowski problem. J. Differ. Geom. 41, 227–246 (1995)

    Article  MathSciNet  Google Scholar 

  39. Lutwak, E., Yang, D., Zhang, G.: Sharp affine \(L_p\) Sobolev inequalities. J. Differ. Geom. 62, 17–38 (2002)

    Article  Google Scholar 

  40. Lutwak, E., Yang, D., Zhang, G.: On the \(L_p\)-Minkowski problem. Trans. Am. Math. Soc. 356, 4359–4370 (2004)

    Article  Google Scholar 

  41. Lutwak, E., Yang, D., Zhang, G.: \(L_p\) dual curvature measures. Adv. Math. 329, 85–132 (2018)

    Article  MathSciNet  Google Scholar 

  42. Minkowski, H.: Allgemeine Lehrsätze über die konvexen Polyeder, Nachr. Ges. Wiess. Göttingen, 189–219 (German) (1897)

  43. Minkowski, H.: Volumen und Oberfläche. Math. Ann. 57, 447–495 (1903). ((German))

    Article  MathSciNet  Google Scholar 

  44. Schneider, R.: Convex Bodies: the Brunn–Minkowski Theory, Encyclopedia of Mathematics and its Applications, vol. 151. Cambridge University Press, Cambridge (2014)

    Google Scholar 

  45. Stancu, A.: The discrete planar \(L_0\)-Minkowski problem. Adv. Math. 167, 160–174 (2002)

    Article  MathSciNet  Google Scholar 

  46. Stancu, A.: On the number of solutions to the discrete two-dimensional \(L_0\)-Minkowski problem. Adv. Math. 180, 290–323 (2003)

    Article  MathSciNet  Google Scholar 

  47. Sun, Y., Long, Y.: The planar Orlicz Minkowski problem in the \(L^1\)-sense. Adv. Math. 281, 1364–1383 (2015)

    Article  MathSciNet  Google Scholar 

  48. Wang, H.: Continuity of the solution to the \(L_p\) Minkowski problem in Gaussian probability space. Acta Math. Sin. (Engl. Ser.), 38, 2253–2264 (2022)

    Article  MathSciNet  Google Scholar 

  49. Wang, H., Fang, N., Zhou, J.: Continuity of the solution to the dual Minkowski problem for negative indices. Proc. Am. Math. Soc. 147, 1299–1312 (2019)

    Article  MathSciNet  Google Scholar 

  50. Wang, H., Fang, N., Zhou, J.: Continuity of the solution to the even logarithmic Minkowski problem in the plane. Sci. China Math. 62, 1419–1428 (2019)

    Article  MathSciNet  Google Scholar 

  51. Wang, H., Lv, Y.: Continuity of the solution to the even \(L_p\) Minkowski problem for \(0<p<1\) in the plane. Intern. J. Math. 31(12), 2050101 (2020)

    Article  Google Scholar 

  52. Wang, T.: The affine Sobolev–Zhang inequality on \(BV({\mathbb{R} }^n)\). Adv. Math. 230, 2457–2473 (2012)

    Article  MathSciNet  Google Scholar 

  53. Wu, Y., Xi, D., Leng, G.: On the discrete Orlicz Minkowski problem. Trans. Am. Math. Soc. 371, 1795–1814 (2019)

    Article  MathSciNet  Google Scholar 

  54. Xi, D., Zhang, Z.: The \(L_p\) Brunn–Minkowski inequality for dual quermassintegrals. Proc. Am. Math. Soc. 150, 3075–3086 (2022)

    Article  Google Scholar 

  55. Xing, S., Ye, D.: On the general dual Orlicz–Minkowski problem. Indiana Univ. Math. J. 69, 621–655 (2020)

    Article  MathSciNet  Google Scholar 

  56. Zhang, G.: The affine Sobolev inequality. J. Differ. Geom. 53, 183–202 (1999)

    Article  MathSciNet  Google Scholar 

  57. Zhang, Z.: The Brunn–Minkowski inequalities of entropy of convex body. Pure Math. 11(7), 1361–1368 (2021). ((In Chinese))

    Article  Google Scholar 

  58. Zhao, Y.: The dual Minkowski problem for negative indices. Calc. Var. Partial Differ. Equ. 56(2), 18 (2017)

    Article  MathSciNet  Google Scholar 

  59. Zhao, Y.: Existence of solution to the even dual Minkowski problem. J. Differ. Geom. 110, 543–572 (2018)

    Article  MathSciNet  Google Scholar 

  60. Zhao, Y.: The \(L_p\) Aleksandrov problem for origin-symmetric polytopes. Proc. Am. Math. Soc. 147, 4477–4492 (2019)

    Article  Google Scholar 

  61. Zhu, B., Xing, S., Ye, D.: The dual Orlicz–Minkowski problem. J. Geom. Anal. 28, 3829–3855 (2018)

    Article  MathSciNet  Google Scholar 

  62. Zhu, B., Zhou, J., Xu, W.: Dual Orlicz–Brunn–Minkowski theory. Adv. Math. 264, 700–725 (2014)

    Article  MathSciNet  Google Scholar 

  63. Zhu, G.: The logarithmic Minkowski problem for polytopes. Adv. Math. 262, 909–931 (2014)

    Article  MathSciNet  Google Scholar 

  64. Zhu, G.: The \(L_p\) Minkowski problem for polytopes for \(0<p<1\). J. Funct. Anal. 269, 1070–1094 (2015)

    Article  MathSciNet  Google Scholar 

  65. Zhu, G.: The centro-affine Minkowski problem for polytopes. J. Differ. Geom. 101, 159–174 (2015)

    Article  MathSciNet  Google Scholar 

  66. Zhu, G.: Continuity of the solution to the \(L_p\) Minkowski problem. Proc. Am. Math. Soc. 145, 379–386 (2017)

    Article  Google Scholar 

  67. Zhu, G.: The \(L_p\) Minkowski problem for polytopes for \(p<0\). Indiana Univ. Math. J. 66, 1333–1350 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors like to thank anonymous referees for comments and suggestions that directly lead to improvement of the early manuscript.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Ji’nan, 250014, Shandong, People’s Republic of China

    Hejun Wang

  2. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

    Hejun Wang & Jiazu Zhou

  3. College of Science, Wuhan University of Science and Technology, Wuhan, 430081, Hubei, People’s Republic of China

    Jiazu Zhou

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  1. Hejun Wang
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Correspondence to Jiazu Zhou.

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Supported in part by NSFC (No.12071378, No.12301066) China Postdoctoral Science Foundation (No.2020M682222) and Natural Science Foundation of Shandong (No.ZR2020QA003, No.ZR2020QA004).

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Wang, H., Zhou, J. Uniqueness and Continuity of the Solution to \(L_p\) Dual Minkowski Problem. Commun. Math. Stat. (2024). https://doi.org/10.1007/s40304-023-00374-2

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