Home On the polar dualities and star dualities of the quasi Lp -intersection bodies
Article
Licensed
Unlicensed Requires Authentication

On the polar dualities and star dualities of the quasi Lp -intersection bodies

  • Yanping Zhou and Weidong Wang EMAIL logo
Published/Copyright: April 24, 2024
Become an author with De Gruyter Brill

Abstract

Yu, Wu and Leng defined the quasi L p -intersection bodies. In this paper, we consider the polar dualities and star dualities of quasi L p -intersection bodies and establish some related inequalities.

MSC 2020: 52A20; 52A40

Award Identifier / Grant number: 11371224

Award Identifier / Grant number: 11901346

Funding statement: Research was partially supported by the Natural Science Foundation of China (Grants No. 11371224 and No. 11901346).

References

[1] G. Berck, Convexity of L p -intersection bodies, Adv. Math. 222 (2009), no. 3, 920–936. 10.1016/j.aim.2009.05.009Search in Google Scholar

[2] R. J. Gardner, A positive answer to the Busemann–Petty problem in three dimensions, Ann. of Math. (2) 140 (1994), no. 2, 435–447. 10.2307/2118606Search in Google Scholar

[3] R. J. Gardner, Intersection bodies and the Busemann–Petty problem, Trans. Amer. Math. Soc. 342 (1994), no. 1, 435–445. 10.1090/S0002-9947-1994-1201126-7Search in Google Scholar

[4] R. J. Gardner, On the Busemann–Petty problem concerning central sections of centrally symmetric convex bodies, Bull. Amer. Math. Soc. (N. S.) 30 (1994), no. 2, 222–226. 10.1090/S0273-0979-1994-00493-8Search in Google Scholar

[5] R. J. Gardner, Geometric Tomography, 2nd ed., Encyclopedia Math. Appl. 58, Cambridge University, New York, 2006. 10.1017/CBO9781107341029Search in Google Scholar

[6] C. Haberl, L p intersection bodies, Adv. Math. 217 (2008), no. 6, 2599–2624. 10.1016/j.aim.2007.11.013Search in Google Scholar

[7] C. Haberl and M. Ludwig, A characterization of L p intersection bodies, Int. Math. Res. Not. IMRN 2006 (2006), Article ID 10548. Search in Google Scholar

[8] N. J. Kalton and A. Koldobsky, Intersection bodies and L p -spaces, Adv. Math. 196 (2005), no. 2, 257–275. 10.1016/j.aim.2004.09.002Search in Google Scholar

[9] A. Koldobsky, Intersection bodies and the Busemann–Petty problem, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 11, 1181–1186. 10.1016/S0764-4442(97)83550-1Search in Google Scholar

[10] A. Koldobsky, Intersection bodies in 𝐑 4 , Adv. Math. 136 (1998), no. 1, 1–14. 10.1006/aima.1998.1718Search in Google Scholar

[11] A. Koldobsky, Intersection bodies, positive definite distributions, and the Busemann–Petty problem, Amer. J. Math. 120 (1998), no. 4, 827–840. 10.1353/ajm.1998.0030Search in Google Scholar

[12] A. Koldobsky, A functional analytic approach to intersection bodies, Geom. Funct. Anal. 10 (2000), no. 6, 1507–1526. 10.1007/PL00001659Search in Google Scholar

[13] F. Lu and G. Leng, On dual quermassintegrals of mixed intersection bodies, J. Math. Anal. Appl. 339 (2008), no. 1, 399–404. 10.1016/j.jmaa.2007.06.033Search in Google Scholar

[14] F. Lu, W. Mao and G. Leng, On star duality of mixed intersection bodies, J. Inequal. Appl. 37 (2007), Article ID 39345. 10.1155/2007/39345Search in Google Scholar

[15] M. Ludwig, Minkowski valuations, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4191–4213. 10.1090/S0002-9947-04-03666-9Search in Google Scholar

[16] M. Ludwig, Intersection bodies and valuations, Amer. J. Math. 128 (2006), no. 6, 1409–1428. 10.1353/ajm.2006.0046Search in Google Scholar

[17] E. Lutwak, Intersection bodies and dual mixed volumes, Adv. Math. 71 (1988), no. 2, 232–261. 10.1016/0001-8708(88)90077-1Search in Google Scholar

[18] M. Moszyńska, Quotient star bodies, intersection bodies, and star duality, J. Math. Anal. Appl. 232 (1999), no. 1, 45–60. 10.1006/jmaa.1998.6238Search in Google Scholar

[19] Y. Pei and W. Wang, A type of Busemann–Petty problems for general Lp-intersection bodies, Wuhan Univ. J. Nat. Sci. 20 (2015), no. 6, 471–475. 10.1007/s11859-015-1121-xSearch in Google Scholar

[20] Z. Shen, Y. Li and W. Wang, L p -dual geominimal surface areas for the general L p -intersection bodies, J. Nonlinear Sci. Appl. 10 (2017), no. 7, 3519–3529. 10.22436/jnsa.010.07.14Search in Google Scholar

[21] W. Wang and Y. Li, General L p -intersection bodies, Taiwanese J. Math. 19 (2015), no. 4, 1247–1259. 10.11650/tjm.19.2015.3493Search in Google Scholar

[22] W. D. Wang and Y. N. Li, Busemann–Petty problems for general L p -intersection bodies, Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 5, 777–786. 10.1007/s10114-015-4273-xSearch in Google Scholar

[23] W. Y. Yu, D. H. Wu and G. S. Leng, Quasi L p -intersection bodies, Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 11, 1937–1948. 10.1007/s10114-007-0958-0Search in Google Scholar

[24] J. Yuan and W.-S. Cheung, L p intersection bodies, J. Math. Anal. Appl. 338 (2008), no. 2, 1431–1439. 10.1016/j.jmaa.2007.06.014Search in Google Scholar

[25] G. Y. Zhang, Intersection bodies and the four-dimensional Busemann–Petty problem, Int. Math. Res. Not. IMRN 1993 (1993), no. 7, 233–240. Search in Google Scholar

[26] G. Y. Zhang, Intersection bodies and the Busemann–Petty inequalities in 𝐑 4 , Ann. of Math. (2) 140 (1994), no. 2, 331–346. 10.2307/2118603Search in Google Scholar

[27] Y. Zhou and S. Wu, Busemann–Petty problems for quasi L p intersection bodies, J. Funct. Spaces (2017), Article ID 3124285. 10.1155/2017/3124285Search in Google Scholar

Received: 2023-09-18
Accepted: 2024-02-12
Published Online: 2024-04-24
Published in Print: 2024-12-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 15.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/gmj-2024-2019/html
Scroll to top button