Browse > Article
http://dx.doi.org/10.4134/BKMS.b160317

ON BONNESEN-STYLE ALEKSANDROV-FENCHEL INEQUALITIES IN ℝn  

Zeng, Chunna (College of Mathematics Science Chongqing Normal University)
Publication Information
Bulletin of the Korean Mathematical Society / v.54, no.3, 2017 , pp. 799-816 More about this Journal
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;
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
연도 인용수 순위
1 T. F. Banchoff and W. F. Pohl, A generalization of the isoperimetric inequality, J. Differential Geom. 6 (1971), 175-213.   DOI
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.   DOI
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 A. Dinghas, Bemerkung zu einer Verscharfung der isoperimetrischen Ungleichung durch H. Hadwiger, Math. Nachr. 1 (1948), 284-286.   DOI
7 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.   DOI
8 V. Diskant, Strengthening of an isoperimetric inequality, Sibirskii Mat. Zh. 14 (1973), 873-877. English translation: Sib. Math. J. 14 (1973), 608-611.
9 V. Diskant, A generalization of Bonnesen's inequalities, Dokl. Math. 14 (1973), 1728-1731.
10 H. Flanders, A proof of Minkowski's inequality for convex curves, Amer. Math. Monthly 75 (1968), no. 6, 581-593.   DOI
11 M. Gage, An isoperimetric inequality with applications to curve shortening, Duke Math. J. 50 (1983), no. 4, 1225-1229.   DOI
12 X. Gao, A new reverse isoperimetric inequality and its stability, Math. Inequal. Appl. 15 (2012), no. 3, 733-743.
13 R. Gardner, Geometric Tomography, Cambridge Univ. Press, New York, 1995.
14 R. Gardner, The Brunn-Minkowski inequality, Minkowski's first inequality, and their duals, J. Math. Anal. Appl. 245 (2000), no. 2, 502-512.   DOI
15 Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer-Verlag Berlin Heidelberg, 1988.
16 R. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. 39 (2002), no. 3, 355-405.   DOI
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.   DOI
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.   DOI
20 L. Gysin, The isoperimetric inequality for nonsimple closed curves, Proc. Amer. Math. Soc. 118 (1993), no. 1, 197-203.   DOI
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.   DOI
26 R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1182-1238.   DOI
27 R. Osserman, Bonnesen-style isoperimetric inequality, Amer. Math. Monthly 86 (1979), no. 1, 1-29.   DOI
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 D. Ren, Topics in Integral Geometry, World Scientific, Sigapore, 1994.
30 S. Pan and H. Xu, Stability of a reverse isoperimetric inequality, J. Math. Anal. Appl. 350 (2009), no. 1, 348-353.   DOI
31 J. R. Sangwine-Yager, Bonnesen-style inequalities for Minkowski relative geometry, Trans. Amer. Math. Soc. 307 (1988), no. 1, 373-382.   DOI
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 R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge Univ. Press, Cambridge, 1993.
34 Y. Xia, On reverse isoperimetric inequalities in two-dimensional space forms and related results, Math. Inequal. Appl. 18 (2015), no. 3, 1025-1032.
35 W. Xu, J. Zhou, and B. Zhu, On containment measure and the mixed isoperimetric inequality, J. Inequal. Appl. 2013 (2103), 540, 11 pp.   DOI
36 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.   DOI
37 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.
38 G. Zhang, Geometric inequalities and inclusion measures of convex bodies, Mathematika 41 (1994), no. 1, 95-116.   DOI
39 G. Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999), no. 1, 183-202.   DOI
40 X.-M. Zhang, Bonnesen-style inequalities and pseudo-perimeters for polygons, J. Geom. 60 (1997), no. 1-2, 188-201.   DOI
41 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.   DOI
42 X.-M. Zhang, Schur-convex functions and isoperimetric inequalities, Proc. Amer. Math. Soc. 126 (1998), no. 2, 461-470.   DOI
43 J. Zhou, Bonnesen-type inequalities, Acta Math. Sin. (Chin. Ser.) 50 (2007), no. 6, 1397-1402.
44 L. A. Santalo, Integral Geometry and Geometric Probability, Reading, MA: Addison- Wesley, 1976.
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.   DOI
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.   DOI
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.