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The \(L_p\) chord Minkowski problem in a critical interval

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Abstract

Chord measures and \(L_p\) chord measures were recently introduced by Lutwak-**-Yang-Zhang by establishing a variational formula regarding a family of fundamental integral geometric invariants called chord integrals. Prescribing the \(L_p\) chord measure is known as the \(L_p\) chord Minkowski problem, which includes the \(L_p\) Minkowski problem heavily studied in the past 2 decades as special cases. In the current work, we solve the \(L_p\) chord Minkowski problem when \(0\le p<1\), without symmetry assumptions.

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Notes

  1. As a comparison, the classical Minkowski problem studies the surface area measure which is also known as the area measure \(S_{n-1}\).

References

  1. Minkowski, H.: Volumen und Oberfläche. Math. Ann. 57(4), 447–495 (1903)

    MathSciNet  Google Scholar 

  2. Aleksandrov, A.: Über die Oberflächenfunktion eines konvexen Körpers. (Bemerkung zur Arbeit “Zur Theorie der gemischten Volumina von konvexen Körpern”). Rec. Math. N.S. [Mat. Sbornik], 6(48):167–174, (1939)

  3. Cheng, S.Y., Yau, S.T.: On the regularity of the solution of the \(n\)-dimensional Minkowski problem. Comm. Pure Appl. Math. 29, 495–516 (1976)

    MathSciNet  Google Scholar 

  4. Pogorelov, A. V.: The Minkowski multidimensional problem. V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto-London, (1978)

  5. Caffarelli, L.A.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. Math. (2) 130(1), 189–213 (1989)

    MathSciNet  Google Scholar 

  6. Caffarelli, L.A.: A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity. Ann. Math. (2) 131(1), 129–134 (1990)

    MathSciNet  Google Scholar 

  7. Caffarelli, L.: Interior \(W^{2, p}\) estimates for solutions of the Monge-Ampère equation. Ann. Math. 2(131), 135–150 (1990)

    Google Scholar 

  8. Guan, B., Guan, P.: Convex hypersurfaces of prescribed curvatures. Ann. Math. 2(156), 655–673 (2002)

    MathSciNet  Google Scholar 

  9. Guan, P., Li, J., Li, Y.: Hypersurfaces of prescribed curvature measure. Duke Math. J. 161, 1927–1942 (2012)

    MathSciNet  Google Scholar 

  10. Guan, P., Ma, X.-N.: The Christoffel-Minkowski problem. I. Convexity of solutions of a Hessian equation. Invent. Math. 151, 553–577 (2003)

    MathSciNet  Google Scholar 

  11. Guan, Pengfei, Ma, **-Nan., Zhou, Feng: The Christofel-Minkowski problem. III. Existence and convexity of admissible solutions. Comm. Pure Appl. Math. 59(9), 1352–1376 (2006)

    MathSciNet  Google Scholar 

  12. Gardner, R.J.: A positive answer to the Busemann-Petty problem in three dimensions. Ann. Math. 2(140), 435–447 (1994)

    MathSciNet  Google Scholar 

  13. Gardner, R.J., Koldobsky, A., Schlumprecht, T.: An analytic solution to the Busemann-Petty problem on sections of convex bodies. Ann. Math. 2(149), 691–703 (1999)

    MathSciNet  Google Scholar 

  14. Lutwak, E.: Intersection bodies and dual mixed volumes. Adv. Math. 71, 232–261 (1988)

    MathSciNet  Google Scholar 

  15. Zhang, G.: A positive solution to the Busemann-Petty problem in \(\mathbb{R} ^4\). Ann. Math. 2(149), 535–543 (1999)

    Google Scholar 

  16. Huang, Y., Lutwak, E., Yang, D., Zhang, G.: Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems. Acta Math. 216(2), 325–388 (2016)

    MathSciNet  Google Scholar 

  17. Böröczky, K.J., Henk, M., Pollehn, H.: Subspace concentration of dual curvature measures of symmetric convex bodies. J. Differ. Geom. 109(3), 411–429 (2018)

    MathSciNet  Google Scholar 

  18. Chen, H., Chen, S., Li, Q.-R.: Variations of a class of Monge-Ampère-type functionals and their applications. Anal. PDE 14(3), 689–716 (2021)

    MathSciNet  Google Scholar 

  19. Chen, C., Huang, Y., Zhao, Y.: Smooth solutions to the \(L_p\) dual Minkowski problem. Math. Ann. 373(3–4), 953–976 (2019)

    MathSciNet  Google Scholar 

  20. Chen, S., Li, Q.-R.: On the planar dual Minkowski problem. Adv. Math. 333, 87–117 (2018)

    MathSciNet  Google Scholar 

  21. Gardner, R.J., Hug, D., Weil, W., **ng, S., Ye, D.: General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski problem I. Calc. Var. Partial Differ. Eqs. 58(1), 12 (2019)

    MathSciNet  Google Scholar 

  22. Henk, M., Pollehn, H.: Necessary subspace concentration conditions for the even dual Minkowski problem. Adv. Math. 323, 114–141 (2018)

    MathSciNet  Google Scholar 

  23. Li, Q.-R., Sheng, W., Wang, X.-J.: Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems. J. Eur. Math. Soc. (JEMS) 22(3), 893–923 (2020)

    MathSciNet  Google Scholar 

  24. Liu, Y., Lu, J.: A flow method for the dual Orlicz-Minkowski problem. Trans. Amer. Math. Soc. 373(8), 5833–5853 (2020)

    MathSciNet  Google Scholar 

  25. Zhao, Y.: Existence of solutions to the even dual Minkowski problem. J. Differ. Geom. 110(3), 543–572 (2018)

    MathSciNet  Google Scholar 

  26. Knüpfer, Hans, Muratov, Cyrill B.: On an isoperimetric problem with a competing nonlocal term I: The planar case. Comm. Pure Appl. Math. 66(7), 1129–1162 (2013)

    MathSciNet  Google Scholar 

  27. Knüpfer, Hans, Muratov, Cyrill B.: On an isoperimetric problem with a competing nonlocal term II: The general case. Comm. Pure Appl. Math. 67(12), 1974–1994 (2014)

    MathSciNet  Google Scholar 

  28. Figalli, A., Fusco, N., Maggi, F., Millot, V., Morini, M.: Isoperimetry and stability properties of balls with respect to nonlocal energies. Comm. Math. Phys. 336(1), 441–507 (2015)

    MathSciNet  Google Scholar 

  29. Haddad, J., Ludwig, M.: Affine fractional \({L}^p\) sobolev inequalities (2022)

  30. Lutwak, E., **, D., Yang, D., Zhang, G.: Chord measure in integral geometry and their Minkowski problems. Comm. Pure Appl. Math., in press

  31. Böröczky, K.J., Lutwak, E., Yang, D., Zhang, G.: The logarithmic Minkowski problem. J. Amer. Math. Soc. 26(3), 831–852 (2013)

    MathSciNet  Google Scholar 

  32. **, D., Yang, D., Zhang, G., Zhao, Y.: The \({L}_p\) chord Minkowski problem. Advanced Nonlinear Studies, in press

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

    MathSciNet  Google Scholar 

  34. Lutwak, E.: The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas. Adv. Math. 118, 244–294 (1996)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  42. 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)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  44. Henk, M., Linke, E.: Cone-volume measures of polytopes. Adv. Math. 253, 50–62 (2014)

    MathSciNet  Google Scholar 

  45. Ludwig, M.: General affine surface areas. Adv. Math. 224, 2346–2360 (2010)

    MathSciNet  Google Scholar 

  46. Ludwig, M., Reitzner, M.: A classification of \({\rm SL}(n)\) invariant valuations. Ann. Math. 2(172), 1219–1267 (2010)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  49. **ong, G.: Extremum problems for the cone volume functional of convex polytopes. Adv. Math. 225, 3214–3228 (2010)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

  53. Bianchi, G., Böröczky, K., Colesanti, A., Yang, D.: The \(L_p\)-Minkowski problem for \(-n<p<1\). Adv. Math. 341, 493–535 (2019)

    MathSciNet  Google Scholar 

  54. Guang, Q., Li, Q.-R., Wang, X.-J.: The \(l_p\)-minkowski problem with super-critical exponents, (2022)

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

    MathSciNet  Google Scholar 

  56. Böröczky, K.J., Lutwak, E., Yang, D., Zhang, G.: The log-Brunn-Minkowski inequality. Adv. Math. 231, 1974–1997 (2012)

    MathSciNet  Google Scholar 

  57. Chen, S., Huang, Y., Li, Q.-R., Liu, J.: The \(L_ p\)-Brunn-Minkowski inequality for \(p<1\). Adv. Math. 368, 107166 (2020)

    MathSciNet  Google Scholar 

  58. Colesanti, A., Livshyts, G., Marsiglietti, A.: On the stability of Brunn-Minkowski type inequalities. J. Funct. Anal. 273(3), 1120–1139 (2017)

    MathSciNet  Google Scholar 

  59. Kolesnikov, A.V., Livshyts, G.: On the Local Version of the Log-Brunn-Minkowski Conjecture and Some New Related Geometric Inequalities. Int. Math. Res. Not. IMRN 18, 14427–14453 (2022)

    MathSciNet  Google Scholar 

  60. Kolesnikov, Alexander, Milman, Emanuel. Local \(L^p\)-Brunn–Minkowski inequalities for \(p<1\). Mem. Amer. Math. Soc., 277(1360), (2022)

  61. Milman, E.: Centro-affine differential geometry and the log-minkowski problem. J. Eur. Math. Soc. (JEMS), accepted

  62. Putterman, E.: Equivalence of the local and global versions of the \(L^p\)-Brunn-Minkowski inequality. J. Funct. Anal. 280(9), 108956 (2021)

    MathSciNet  Google Scholar 

  63. Saroglou, C.: Remarks on the conjectured log-Brunn-Minkowski inequality. Geom. Dedicata 177, 353–365 (2015)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  65. Chen, S., Li, Q.-R., Zhu, G.: On the \(L_p\) Monge-Ampère equation. J. Differ. Eqs. 263(8), 4997–5011 (2017)

    Google Scholar 

  66. Chen, S., Li, Q.-R., Zhu, G.: The logarithmic Minkowski problem for non-symmetric measures. Trans. Amer. Math. Soc. 371(4), 2623–2641 (2019)

    MathSciNet  Google Scholar 

  67. Schneider, R.: Convex bodies: the Brunn-Minkowski theory, volume 151 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, expanded edition (2014)

  68. Santaló, L. A.: Integral geometry and geometric probability. Cambridge Mathematical Library. Cambridge University Press, Cambridge, second edition, 2004. With a foreword by Mark Kac

  69. Ren, D.: Topics in integral geometry, volume 19 of Series in Pure Mathematics. World Scientific Publishing Co., Inc., River Edge, NJ, 1994. Translated from the Chinese and revised by the author, With forewords by Shiing Shen Chern and Chuan-Chih Hsiung

  70. Zhang, G.: Dual kinematic formulas. Trans. Amer. Math. Soc. 351, 985–995 (1999)

    MathSciNet  Google Scholar 

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Acknowledgements

The authors are extremely grateful to the referees for their many valuable comments and suggestions. Research of Guo was supported, in part, by NSFC Grants 12126319 and 12126368. Research of ** was supported, in part, by NSFC Grant 12071277 and STCSM Grant 20JC1412600. Research of Zhao was supported, in part, by NSF Grant DMS–2132330.

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Authors and Affiliations

  1. College of Mathematics and Information Science, Henan Normal University, Henan, 453007, China

    Lujun Guo

  2. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Dongmeng **

  3. Department of Mathematics, Syracuse University, Syracuse, 13244, NY, USA

    Yiming Zhao

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  1. Lujun Guo
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  2. Dongmeng **
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  3. Yiming Zhao
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Correspondence to Dongmeng **.

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Guo, L., **, D. & Zhao, Y. The \(L_p\) chord Minkowski problem in a critical interval. Math. Ann. 389, 3123–3162 (2024). https://doi.org/10.1007/s00208-023-02721-8

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