# A STABILITY RESULT FOR P-CENTROID BODIES

• Guo, Lujun (Department of Mathematics and Information Science Henan Normal University) ;
• Leng, Gangsong (Department of Mathematics Shanghai University) ;
• Lin, Youjiang (Institute of Mathematics and Statistics Chongqing Technology and Business University)
• Received : 2016.11.02
• Accepted : 2017.06.08
• Published : 2018.01.31

#### Abstract

In this paper, we prove a stability result for p-centroid bodies with respect to the Hausdorff distance. As its application, we show that the symmetric convex body is determined by its p-centroid body.

#### Acknowledgement

Supported by : National Natural Science Foundation of China, CQCSTC, Chongqing Municipal Education Commission

#### References

1. S. Campi and P. Gronchi, The $L^p$-Busemann-Petty centroid inequlity, Adv. Math. 167 (2002), 128-141. https://doi.org/10.1006/aima.2001.2036
2. S. Campi and P. Gronchi, On the reverse $L_p$-Busemann-Petty centroid inequality, Mathematika 49 (2002), no. 1-2, 1-11. https://doi.org/10.1112/S0025579300016004
3. I. Fary and L. Redei, Der zentralsymmetrische Kern und die zentralsymmetrische Hulle von konvexen Korpern, Math. Ann. 122 (1950), 205-220. https://doi.org/10.1007/BF01342966
4. B. Fleury, O. Guedon, and G. Paouris, A stability result for mean width of $L_p$-centroid bodies, Adv. Math. 214 (2007), no. 2, 865-877. https://doi.org/10.1016/j.aim.2007.03.008
5. R. J. Gardner and A. Giannopoulos, P-cross-section bodies, Indiana Univ. Math. J. 48 (1999), no. 2, 593-613.
6. H. Groemer, Geometric Applications of Fourier Series and Spherical Harmonics, Cam-bridge University Press, New York, 1996.
7. P. M. Gruber, Convex and Discrete Geometry, Grundlehren Math. Wiss., vol. 336, Springer, Berlin, 2007.
8. L. Guo and G. Leng, Determination of star bodies from p-centroid bodies, Proc. Indian Acad. Sci. Math. Sci. 123 (2013), no. 4 , 577-586. https://doi.org/10.1007/s12044-013-0143-6
9. L. Guo and G. Leng, Stable determination of convex bodies from centroid bodies, Houston J. Math. 40 (2014), no. 2, 395-406.
10. A. Koldobsky, Common subspaces of $L_p$-spaces, Proc. Amer. Math. Soc. 122 (1994), no. 1, 207-212. https://doi.org/10.2307/2160862
11. E. Lutwak, On some affine isoperimetric inequalities, J. Differential Geom. 23 (1986), no. 1, 1-13. https://doi.org/10.4310/jdg/1214439900
12. E. Lutwak, Centroid bodies and dual mixed volumes, Proc. London Math. Soc. 60 (1990), no. 2, 365-391.
13. E. Lutwak, The Brunn-Minkowski-Firey theory I: Mixed volumes and the Minkowski problem , J. Differential Geom. 38 (1993), no. 1, 131-150. https://doi.org/10.4310/jdg/1214454097
14. E. Lutwak, The Brunn-Minkowski-Firey theory II: Affine and geominimal surface areas, Adv. Math. 118 (1996), no. 2, 244-294. https://doi.org/10.1006/aima.1996.0022
15. E. Lutwak, D. Yang, and G. Zhang, $L_p$ affine isoperimetric inequalities, J. Differential Geom. 56 (2000), no. 1, 111-132. https://doi.org/10.4310/jdg/1090347527
16. E. Lutwak, D. Yang, and G. Zhang, Sharp affine $L_p$ Sobolev inequalities, J. Differential Geom. 62 (2002), no. 1, 17-38. https://doi.org/10.4310/jdg/1090425527
17. E. Lutwak, D. Yang, and G. Zhang, Orlicz centroid bodies, J. Differential Geom. 84 (2010), no. 2, 365-387. https://doi.org/10.4310/jdg/1274707317
18. E. Lutwak and G. Zhang, Blaschke-Santalo inequalities, J. Differential Geom. 47 (1997), no. 1, 1-16. https://doi.org/10.4310/jdg/1214460036
19. V. D. Milman and A. Pajor, Isotropic position and inertia ellipsoid and zonoids of the unit ball of a normed n-dimensional space, Geometric aspects of functional analysis (19878), 64-104, Lecture Notes in Math., 1376, Springer, Berlin, 1989.
20. G. Paouris, Concentration of mass on isotropic convex bodies, C. R. Math. Acad. Sci. Paris 342 (2006), no. 3, 179-182. https://doi.org/10.1016/j.crma.2005.11.018
21. G. Paouris, Small ball probability estimates for log-concave measures, Trans. Amer. Math. Soc. 364 (2012), no. 1, 287-308. https://doi.org/10.1090/S0002-9947-2011-05411-5
22. C. M. Petty, Centroid surfaces, Pacific J. Math. 11 (1961), 1535-1547. https://doi.org/10.2140/pjm.1961.11.1535
23. R. Schneider, Convex bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathe-matics and its Applications, Vol. 44, Cambridge University Press, Cambridge, 1993.
24. G. Zhang, Centered bodies and dual mixed volumes, Trans. Amer. Math. Soc. 345 (1994), no. 2, 777-801. https://doi.org/10.1090/S0002-9947-1994-1254193-9
25. G. Zhu, The Orlicz centroid inequality for star bodies, Adv. Appl. Math. 48 (2012), no. 2, 432-445. https://doi.org/10.1016/j.aam.2011.11.001
26. G. Zhu, The logarithmic Minkowski problem for polytopes, Adv. Math. 262 (2014), 909-931. https://doi.org/10.1016/j.aim.2014.06.004
27. G. Zhu, The centro-affine Minkowski problem for polytopes, J. Differential Geom. 101 (2015), no. 1, 159-174. https://doi.org/10.4310/jdg/1433975485