# ON BONNESEN-STYLE ALEKSANDROV-FENCHEL INEQUALITIES IN ℝn

• Published : 2017.05.31
• 42 11

#### Abstract

In this paper, we investigate the Bonnesen-style Aleksandrov-Fenchel inequalities in ${\mathbb{R}}^n$, which are the generalization of known Bonnesen-style inequalities. We first define the i-th symmetric mixed homothetic deficit ${\Delta}_i(K,L)$ and its special case, the i-th Aleksandrov-Fenchel isoperimetric deficit ${\Delta}_i(K)$. Secondly, we obtain some lower bounds of (n - 1)-th Aleksandrov Fenchel isoperimetric deficit ${\Delta}_{n-1}(K)$. Theorem 4 strengthens Groemer's result. As direct consequences, the stronger isoperimetric inequalities are established when n = 2 and n = 3. Finally, the reverse Bonnesen-style Aleksandrov-Fenchel inequalities are obtained. As a consequence, the new reverse Bonnesen-style inequality is obtained.

#### Keywords

mixed volume;isoperimetric inequality;Bonnesen-style inequality;Aleksandrov-Fenchel inequality

#### References

1. T. F. Banchoff and W. F. Pohl, A generalization of the isoperimetric inequality, J. Differential Geom. 6 (1971), 175-213. https://doi.org/10.4310/jdg/1214430403
2. W. Blaschke, Vorlesungen uber Intergralgeometrie, 3rd ed. Deutsch. VerlagWiss., Berlin, 1955.
3. J. Bokowski and E. Heil, Integral representation of quermassintegrals and Bonnesenstyle inequalities, Arch. Math. 47 (1986), no. 1, 79-89. https://doi.org/10.1007/BF01202503
4. T. Bonnesen, Les problems des isoperimetres et des isepiphanes, Paris, 1929.
5. T. Bonnesen and W. Fenchel, Theorie der konvexen Koeper, 2nd ed., Berlin-Heidelberg-New York, 1974.
6. Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer-Verlag Berlin Heidelberg, 1988.
7. A. Dinghas, Bemerkung zu einer Verscharfung der isoperimetrischen Ungleichung durch H. Hadwiger, Math. Nachr. 1 (1948), 284-286. https://doi.org/10.1002/mana.19480010503
8. V. Diskant, Bounds for the discrepancy between convex bodies in terms of the isoperimetric difference, Sibirskii Mat. Zh. 13 (1972), 767-772. English translation: Siberian Math. J. 13 (1973), 529-532. https://doi.org/10.1007/BF00971045
9. V. Diskant, Strengthening of an isoperimetric inequality, Sibirskii Mat. Zh. 14 (1973), 873-877. English translation: Sib. Math. J. 14 (1973), 608-611.
10. V. Diskant, A generalization of Bonnesen's inequalities, Dokl. Math. 14 (1973), 1728-1731.
11. H. Flanders, A proof of Minkowski's inequality for convex curves, Amer. Math. Monthly 75 (1968), no. 6, 581-593. https://doi.org/10.2307/2313773
12. M. Gage, An isoperimetric inequality with applications to curve shortening, Duke Math. J. 50 (1983), no. 4, 1225-1229. https://doi.org/10.1215/S0012-7094-83-05052-4
13. X. Gao, A new reverse isoperimetric inequality and its stability, Math. Inequal. Appl. 15 (2012), no. 3, 733-743.
14. R. Gardner, Geometric Tomography, Cambridge Univ. Press, New York, 1995.
15. R. Gardner, The Brunn-Minkowski inequality, Minkowski's first inequality, and their duals, J. Math. Anal. Appl. 245 (2000), no. 2, 502-512. https://doi.org/10.1006/jmaa.2000.6774
16. R. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. 39 (2002), no. 3, 355-405. https://doi.org/10.1090/S0273-0979-02-00941-2
17. M. Green and S. Osher, Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves, Asian J. Math. 3 (1999), no. 3, 659-676. https://doi.org/10.4310/AJM.1999.v3.n3.a5
18. H. Groemer, Geometric applications of Fourier series and spherical harmonics, Encyclo-pedia of Mathematics and its Applications, 61. Cambridge University Press, Cambridge, 1996.
19. H. Groemer and R. Schneider, Stability estimates for some geometric inequalities, Bull. Lond. Math. Soc. 23 (1991), no. 1, 67-74. https://doi.org/10.1112/blms/23.1.67
20. L. Gysin, The isoperimetric inequality for nonsimple closed curves, Proc. Amer. Math. Soc. 118 (1993), no. 1, 197-203. https://doi.org/10.1090/S0002-9939-1993-1079698-X
21. H. Hadwiger, Die isoperimetrische Ungleichung in Raum, Elem. Math. 3 (1948), 25-38.
22. H. Hadwiger, Kurze Herleitung einer verscharften isoperimetrischen Ungleichung fur konvexe Korper, Rev. Fac. Sci. Univ. Istanbul, Ser. A 14 (1949), 1-6.
23. H. Hadwiger, Vorlesungen uber Inhalt, Oberflache und Isoperimetrie, Springer, Berlin, 1957.
24. W. Y. Hsiang, An elementary proof of the isoperimetric problem, Chinese Ann. Math. 23 (2002), no. 1, 7-12.
25. D. Klain, Bonnesen-type inequalities for surfaces of constant curvature, Adv. in Appl. Math. 39 (2007), no. 2, 143-154. https://doi.org/10.1016/j.aam.2006.11.004
26. R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1182-1238. https://doi.org/10.1090/S0002-9904-1978-14553-4
27. R. Osserman, Bonnesen-style isoperimetric inequality, Amer. Math. Monthly 86 (1979), no. 1, 1-29. https://doi.org/10.2307/2320297
28. S. Pan, X. Tang, and X. Wang, A refined reverse isoperimetric inequality in the plane, Math. Inequal. Appl. 13 (2010), no. 2, 329-338.
29. S. Pan and H. Xu, Stability of a reverse isoperimetric inequality, J. Math. Anal. Appl. 350 (2009), no. 1, 348-353. https://doi.org/10.1016/j.jmaa.2008.09.047
30. D. Ren, Topics in Integral Geometry, World Scientific, Sigapore, 1994.
31. J. R. Sangwine-Yager, Bonnesen-style inequalities for Minkowski relative geometry, Trans. Amer. Math. Soc. 307 (1988), no. 1, 373-382. https://doi.org/10.1090/S0002-9947-1988-0936821-5
32. J. R. Sangwine-Yager, Mixe Volumes, Handbook of Covex Geometry, Vol. A, 43-71, Edited by P. Gruber & J. Wills, North-Holland, 1993.
33. L. A. Santalo, Integral Geometry and Geometric Probability, Reading, MA: Addison- Wesley, 1976.
34. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge Univ. Press, Cambridge, 1993.
35. Y. Xia, On reverse isoperimetric inequalities in two-dimensional space forms and related results, Math. Inequal. Appl. 18 (2015), no. 3, 1025-1032.
36. W. Xu, J. Zhou, and B. Zhu, On containment measure and the mixed isoperimetric inequality, J. Inequal. Appl. 2013 (2103), 540, 11 pp. https://doi.org/10.1186/1029-242X-2013-11
37. C. Zeng, L. Ma, J. Zhou, and F. Chen, The Bonnesen isoperimetric inequality in a surface of constant curvature, Sci. China Math. 55 (2012), no. 9, 1913-1919. https://doi.org/10.1007/s11425-012-4405-z
38. C. Zeng, J. Zhou, and S. Yue, A symmetric mixed isoperimetric inequality for two planar convex domains, Acta Math. Sinica 55 (2012), no. 2, 355-362.
39. G. Zhang, Geometric inequalities and inclusion measures of convex bodies, Mathematika 41 (1994), no. 1, 95-116. https://doi.org/10.1112/S0025579300007208
40. G. Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999), no. 1, 183-202. https://doi.org/10.4310/jdg/1214425451
41. X.-M. Zhang, Bonnesen-style inequalities and pseudo-perimeters for polygons, J. Geom. 60 (1997), no. 1-2, 188-201. https://doi.org/10.1007/BF01252226
42. X.-M. Zhang, Schur-convex functions and isoperimetric inequalities, Proc. Amer. Math. Soc. 126 (1998), no. 2, 461-470. https://doi.org/10.1090/S0002-9939-98-04151-3
43. J. Zhou, Bonnesen-type inequalities, Acta Math. Sin. (Chin. Ser.) 50 (2007), no. 6, 1397-1402.
44. J. Zhou and F. Chen, The Bonnesen-type inequality in a plane of constant cuvature, J. Korean Math. Soc. 44 (2007), no. 6, 1363-1372. https://doi.org/10.4134/JKMS.2007.44.6.1363
45. J. Zhou, Y. Du, and F. Cheng, Some Bonnesen-style inequalities for higher dimensions, Acta Math. Sin. (Engl. Ser.) 28 (2012), no. 12, 2561-2568. https://doi.org/10.1007/s10114-012-9657-6
46. J. Zhou and D. Ren, Geometric inequalities from the viewpoint of integral geometry, Acta Math. Sci. Ser. A Chin. Ed. 30 (2010), no. 5, 1322-1339.
47. J. Zhou, Y. Xia, and C. Zeng, Some new Bonnesen-style inequalities, J. Korean Math. Soc. 48 (2011), no. 2, 421-430. https://doi.org/10.4134/JKMS.2011.48.2.421
48. J. Zhou, C. Zhou, and F. Ma, Isoperimetric deficit upper limit of a planar convex set, Rend. Circ. Mat. Palermo (2) 81 (2009), 363-367.

#### Acknowledgement

Supported by : Chinese Scholarship Council, Chongqing Educational Committee, CQNU