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Geometry of log-concave functions: the \(L_p\) Asplund sum and the \(L_{p}\) Minkowski problem

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Abstract

The aim of this paper is to develop a basic framework of the \(L_p\) theory for the geometry of log-concave functions, which can be viewed as a functional “lifting” of the \(L_p\) Brunn-Minkowski theory for convex bodies. To fulfill this goal, by combining the \(L_p\) Asplund sum of log-concave functions for all \(p>1\) and the total mass, we obtain a Prékopa-Leindler type inequality and propose a definition for the first variation of the total mass in the \(L_p\) setting. Based on these, we further establish an \(L_p\) Minkowski type inequality related to the first variation of the total mass and derive a variational formula which motivates the definition of our \(L_p\) surface area measure for log-concave functions. Consequently, the \(L_p\) Minkowski problem for log-concave functions, which aims to characterize the \(L_p\) surface area measure for log-concave functions, is introduced. The existence of solutions to the \(L_p\) Minkowski problem for log-concave functions is obtained for \(p>1\) under some mild conditions on the pre-given Borel measures.

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References

  1. Alonso-Gutiérrez, D.: A reverse Rogers-Shephard inequality for log-concave functions. J. Geom. Anal. 29, 299–315 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alonso-Gutiérrez, D., Artstein-Avidan, S., Merino, B.G., Jiménez, C.H., Villa, R.: Rogers-Shephard and local Loomis-Whitney type inequalities. Math. Ann. 374, 1719–1771 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alonso-Gutiérrez, D., Bernués, J., Merino, B.G.: An extension of Berwald’s inequality and its relation to Zhang’s inequality. J. Math. Anal. Appl. 486, 123875 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alonso-Gutiérrez, D., Bernués, J., Merino, B.G.: Zhang’s inequality for log-concave functions. In: Klartag, B., Milman, E. (eds.) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 2256, pp. 29–48. Springer, Cham (2020)

    Chapter  Google Scholar 

  5. Alonso-Gutiérrez, D., Merino, B.G., Jiménez, C.H., Villa, R.: Rogers-Shephard inequality for log-concave functions. J. Funct. Anal. 271, 3269–3299 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Alonso-Gutiérrez, D., Merino, B.G., Jiménez, C.H., Villa, R.: John’s ellipsoid and the integral ratio of a log-concave function. J. Geom. Anal. 28, 1182–1201 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Artstein-Avidan, S., Florentin, D.I., Segal, A.: Functional Brunn-Minkowski inequalities induced by polarity. Adv. Math. 364, 107006 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  8. Artstein-Avidan, S., Klartag, B., Milman, V.D.: The Santaló point of a function, and a functional form of Santaló inequality. Mathematika 51, 33–48 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Artstein-Avidan, S., Klartag, B., Schütt, C., Werner, E.M.: Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality. J. Funct. Anal. 262, 4181–4204 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Artstein-Avidan, S., Milman, V.D.: The concept of duality in convex analysis, and the characterization of the Legendre transform. Ann. Math. 169, 661–674 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Artstein-Avidan, S., Rubinstein, Y.A.: Differential analysis of polarity: Polar Hamilton-Jacobi, conservation laws, and Monge-Ampère equations. J. Anal. Math. 132, 133–156 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Artstein-Avidan, S., Slomka, B.A.: A note on Santaló inequality for the polarity transform and its reverse. Proc. Am. Math. Soc. 143, 1693–1704 (2015)

    Article  MATH  Google Scholar 

  13. Artstein-Avidan, S., Slomka, B.A.: Functional covering numbers. J. Geom. Anal. 31, 1039–1072 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ball, K.: Logarithmically concave functions and sections of convex sets in \({\mathbb{R}}^n\). Studia Math. 1, 69–84 (1988)

    Article  MATH  Google Scholar 

  15. Barthe, F., Böröczky, K.J., Fradelizi, M.: Stability of the functional forms of the Blaschke-Santaló inequality. Monatsh. Math. 173, 135–159 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Berman, R.J., Berndtsson, B.: Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties. Ann. Fac. Sci. Toulouse Math. 22, 649–711 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bobkov, S.: Isoperimetric and analytic inequalites for log-concave probability measures. Ann. Probab. 27, 1903–1921 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Borwein, J.M., Vanderwerff, J.D.: Convex Functions: Constructions, Characterizations, and Counterexamples, Encyclopedia of Mathematics and Its Applications, vol. 109. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  20. Brascamp, H., Lieb, E.: On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to diffusion equation. J. Funct. Anal. 22, 366–389 (1976)

    Article  MATH  Google Scholar 

  21. Caglar, U., Fradelizi, M., Guédon, O., Lehec, J., Schütt, C., Werner, E.M.: Functional versions of \(L_p\)-affine surface area and entropy inequalities. Int. Math. Res. Not. 2016, 1223–1250 (2016)

    Article  MATH  Google Scholar 

  22. Caglar, U., Werner, E.M.: Divergence for \(s\)-concave and log concave functions. Adv. Math. 257, 219–247 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Caglar, U., Werner, E.M.: Mixed \(f\)-divergence and inequalities for log-concave functions. Proc. Lond. Math. Soc. 110, 271–290 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Caglar, U., Ye, D.: Affine isoperimetric inequalities in the functional Orlicz-Brunn-Minkowski theory. Adv. Appl. Math. 81, 78–114 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Chou, K., Wang, X.: The \(L_p\)-Minkowski problem and the Minkowski problem in centro-affine geometry. Adv. Math. 205, 33–83 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Colesanti, A.: Functional inequalities related to the Rogers-Shephard inequality. Mathematika 53, 81–101 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Colesanti, A.: Log-concave functions. In: Convexity and Concentration, New York, (2017)

  29. 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  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Dubuc, S.: Critères de convexité et inégalités intégrales. Ann. Inst. Fourier (Grenoble) 27, 135–165 (1977)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Firey, W.J.: \(p\)-means of convex bodies. Math. Scand. 10, 17–24 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  34. Fradelizi, M., Meyer, M.: Some functional forms of Blaschke-Santaló inequality. Math. Z. 256, 379–395 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  35. Fradelizi, M., Meyer, M.: Some functional inverse Santaló inequality. Adv. Math. 218, 1430–1452 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Gardner, R.J.: The Brunn-Minkowski Inequality. Bull. Amer. Math. Soc. (N.S.) 39, 355–405 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  37. 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, Art. 12, 35 pp. (2019)

  38. Huang, Y., Liu, J., Wang, X.: 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  MATH  Google Scholar 

  39. 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  MATH  Google Scholar 

  40. Jian, H., Lu, J., Wang, X.: Nonuniqueness of solutions to the \(L_p\)-Minkowski problem. Adv. Math. 281, 845–856 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  41. Jian, H., Lu, J., Zhu, G.: Mirror symmetric solutions to the centro-affine Minkowski problem. Calc. Var. Partial Differ. Equ. 55, 1–22 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Klartag, B.: Logarithmically-concave moment measures In: Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 2116, pp. 231–260 (2014)

  43. Klartag, B., Milman, V.D.: Geometry of log-concave functions and measures. Geom. Dedicata 112, 169–182 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  44. Leindler, L.: On a certain converse of Hölder inequality II. Acta Sci. Math. (Szeged) 33, 217–223 (1972)

    MathSciNet  MATH  Google Scholar 

  45. Li, B., Schütt, C., Werner, E.M.: Floating functions. Israel J. Math. 231, 181–210 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  46. Li, B., Schütt, C., Werner, E.M.: The Löwner function of a log-concave function. J. Geom. Anal. 31, 423–456 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  47. Lin, Y.: Affine Orlicz Pólya-Szegö principle for log-concave functions. J. Funct. Anal. 273, 3295–3326 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  50. 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  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  52. Milman, V.D., Rotem, L.: Mixed integrals and related inequalities. J. Funct. Anal. 264, 570–604 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  53. Minkowski, H.: Allgemeine Lehrsätze über die convexen polyeder. Nachr. Ges. Wiss. Göttingen 198–219 (1897)

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

    Article  MathSciNet  MATH  Google Scholar 

  55. Prékopa, A.: Logarithmic concave measures with applications to stochastic programming. Acta Sci. Math. (Szeged) 32, 301–316 (1971)

    MathSciNet  MATH  Google Scholar 

  56. Prékopa, A.: On logarithmic concave measures and functions. Acta Sci. Math. (Szeged) 34, 335–343 (1973)

    MathSciNet  MATH  Google Scholar 

  57. Prékopa, A.: New proof for the basic theorem of logconcave measures (Hungarian). Alkalmaz. Mat. Lapok 1, 385–389 (1975)

    MathSciNet  MATH  Google Scholar 

  58. Rockafellar, T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

  59. Rotem, L.: Support functions and mean width for \(\alpha \)-concave functions. Adv. Math. 243, 168–186 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  60. Rotem, L.: Surface area measures of log-concave functions, J. Anal. Math., (2021), in press

  61. Rotem, L.: A Riesz representation theorem for log-concave functions, arXiv:2105.09168

  62. Santambrogio, F.: Dealing with moment measures via entropy and optimal transport. J. Funct. Anal. 271, 418–436 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  63. Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge (2014)

    MATH  Google Scholar 

  64. Umanskiy, V.: On solvability of two-dimensional \(L_p\)-Minkowski problem. Adv. Math. 180, 176–186 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  66. Zhu, B., Hong, H., Ye, D.: The Orlicz-Petty bodies. Int. Math. Res. Not. 2018, 4356–4403 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The research of NF was supported in part by NSFC (No.12001291), China. The research of DY was supported by a NSERC grant, Canada. The authors are greatly indebted to the reviewer for many valuable comments.

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Authors and Affiliations

  1. School of Mathematics, Hunan University, Changsha, Hunan, 410082, China

    Niufa Fang

  2. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G2G1, Canada

    Sudan Xing

  3. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C5S7, Canada

    Deping Ye

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  1. Niufa Fang
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  2. Sudan Xing
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  3. Deping Ye
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Correspondence to Sudan Xing.

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Communicated by N. S. Trudinger.

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Fang, N., Xing, S. & Ye, D. Geometry of log-concave functions: the \(L_p\) Asplund sum and the \(L_{p}\) Minkowski problem. Calc. Var. 61, 45 (2022). https://doi.org/10.1007/s00526-021-02155-7

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