关于广义L_p-混合投影体与广义L_p-混合质心体的单调不等式（英文）
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摘要
本文研究了广义L_p-混合投影体及广义L_p-混合质心体的单调性问题.利用解析不等式,获得了广义L_p-混合投影体与广义L_p-混合质心体的均质积分与对偶均质积分形式的单调不等式,推广了L_p-投影体及L_p-质心体的体积形式的单调性.
In this paper, we study the problem of the monotonicity on general L_p-mixed projection bodies and general L_p-mixed centroid bodies. By using analytic inequality theory, some monotonic inequalities of quermassintegrals and dual quermassintegrals for general L_p-mixed projection bodies and general L_p-mixed centroid bodies are obtained, which generalizes the problem of the monotonicity for the form of volume on L_p-projection bodies and L_p-centroid bodies.
In this paper, we study the problem of the monotonicity on general L_p-mixed projection bodies and general L_p-mixed centroid bodies. By using analytic inequality theory, some monotonic inequalities of quermassintegrals and dual quermassintegrals for general L_p-mixed projection bodies and general L_p-mixed centroid bodies are obtained, which generalizes the problem of the monotonicity for the form of volume on L_p-projection bodies and L_p-centroid bodies.
引文
[1] Gardner R J. Geometric tomography(2nd ed.)[M]. Cambridge, UK:Cambridge Univ. Press, 2006.
[2] Schneider R. Convex bodies:the Brunn-Minkowski theory(2nd ed.)[M]. Cambridge:Cambridge University Press, 2014.
[3] Lutwak E, Zhang Gaoyong. Blaschke-Santalóinequalities[J]. J. Diff. Geom., 1997, 47(1):1-16.
[4] Lutwak E, Yang D, Zhang Gaoyong. L_p affine isoperimetric inequalities[J]. J. Diff. Geom., 2000,56(1):111-132.
[5] Wang Weidong. On reverses of the L_p-Busemann-Petty centroid inequality and its applications[J].Wuhan Univ. J. Nat. Sci., 2010, 15(4):292-296.
[6] Wang Weidong, Leng Gangsong. Inequalities relating to L_p-version of Petty's conjectured projection inequality[J]. Appl. Math. Mech., 2007, 28(2):269-276.
[7] Wang Weidong, Leng Gangsong. On the monotonicity of Lp-centroid body[J]. J. Sys. Sci. Math.Scis., 2008, 28(2):154-162(in Chinese).
[8] Wang Weidong, Leng Gangsong. Some affine isoperimetric inequalities associated with Lp-affine surface area[J]. Houston J. Math., 2008, 34(2):443-453.
[9] Wang Weidong, Leng Gangsong. On the L_p-version of the Petty's conjectured projection inequality and applications[J]. Taiwan. J. Math., 2008, 12(5):1067-1086.
[10] Wang Weidong, Leng Gangsong. Inequalities of the quermassintegrals for the Lp-projection body and the Lp-centroid body[J]. Acta Math. Sci., 2010, 30B(1):359-368.
[11] Wang Weidong, Leng Gangsong, Lu Fenghong. On Brunn-Minkowski inequality for the quermassintegrals and dual quermassintegrals of Lp-projection bodies[J]. Chinese Math. Ann., 2008, 29A(2):209-220(in Chinese).
[12] Wang Weidong, Lu Fenghong, Leng Gangsong. A type of monotonicity on the Lp centroid body and Lp projection body[J]. Math. Inequal. Appl., 2005, 8(4):735-742.
[13] Wang Weidong, Lu Fenghong, Leng Gangsong. On monotonicity properties of the Lp-centroid bodies[J]. Math. Inequal. Appl., 2013, 16(3):645-655.
[14] Wang Weidong, Wei Daijun, Xiang Yu. On monotony for the L_p-projection body[J]. Chinese Adv.Math., 2008, 37(6):690-700(in Chinese).
[15] Wang Weidong, Wei Daijun, Xiang Yu. On reverses of the L_p-Petty projection inequality[J]. Chin.Quart. J. Math., 2009, 24(4):491-498.
[16] Ludwig M. Minkowski valuations[J]. Trans. Amer. Math. Soc., 2005, 357(10):4191-4213.
[17] Wang Weidong, Wan Xiaoyan. Shephard type problems for general Lp-projection bodies[J]. Taiwan.J. Math., 2012, 16(5):1749-1762.
[18] Wang Weidong, Feng Yibin. A general Lp-version of Petty's affine projection inequality[J]. Taiwan.J. Math., 2013, 17(2):517-528.
[19] Wang Weidong, Wang Jianye. Extremum of geometric functionals involving general Lp-projection bodies[J]. J. Inequal. Appl., 2016, 2016:1-16.
[20] Feng Yibin, Wang Weidong, Lu Fenghong. Some inequalities on general L_p-centroid bodies[J].Math. Inequal. Appl., 2015, 18(1):39-49.
[21] Wang Weidong, Leng Gangsong. The Petty projection inequality for L_p-mixed projection bodies[J]. Acta Math. Sinica(English Series), 2007, 23(8):1485-1494.
[22] Feng Yibin, Wang Weidong. The Shephard type problems and monotonicity for L_p-mixed centroid body[J]. Indian J. Pure Appl. Math., 2014, 45(3):265-283.
[23] Liu Lijuan, Wang Wei, He Binwu. Fourier transform and L_p-mixed projection bodies[J]. Bull.Korean Math. Soc., 2010, 47(5):1011-1023.
[24] Ma Tongyi. On Lp-mixed centroid bodies and dual L_p-mixed centroid bodies[J]. Acta Math. Sinica(Chinese Series), 2010, 53(2):301-314.
[25] Wang Weidong, Wan Xiaoyan. L_p-mixed projection bodies and Lp-mixed quermassintegrals[J]. J.Math. Inequal., 2014, 8(4):879-888.
[26] Wan Xiaoyan, Wang Weidong. Petty projection inequalities for the general L_p-mixed projection bodies[J]. Wuhan Univ. J. Nat. Sci.,2012,17(3):190-194.
[27] Firey W J. p-means of convex bodies[J]. Math. Scand., 1962, 10:17-24.
[28] Lutwak E. The Brunn-Minkowski-Firey theory II:affine and geominimal surface areas[J]. Adv.Math., 1996, 118(2):244-294.
[29] Firey W J. Mean cross-section measures of harmonic means of convex bodies[J]. Pacific J. Math.,1961, 11(4):1263-1266.
[30] Lutwak E. The Brunn-Minkowski-Firey theory I:mixed volumes and the Minkowski problem[J]. J.Diff. Geom., 1993, 38(1):131-150.
[31] Lutwak E. Dual mixed volumes[J]. Pacific J. Math., 1975, 58(2):531-538.
[32] Wang Weidong, Leng Gangsong. L_p-dual mixed quermassintegrals[J]. Indian J. Pure Appl. Math.,2005, 36(4):177-188.
[2] Schneider R. Convex bodies:the Brunn-Minkowski theory(2nd ed.)[M]. Cambridge:Cambridge University Press, 2014.
[3] Lutwak E, Zhang Gaoyong. Blaschke-Santalóinequalities[J]. J. Diff. Geom., 1997, 47(1):1-16.
[4] Lutwak E, Yang D, Zhang Gaoyong. L_p affine isoperimetric inequalities[J]. J. Diff. Geom., 2000,56(1):111-132.
[5] Wang Weidong. On reverses of the L_p-Busemann-Petty centroid inequality and its applications[J].Wuhan Univ. J. Nat. Sci., 2010, 15(4):292-296.
[6] Wang Weidong, Leng Gangsong. Inequalities relating to L_p-version of Petty's conjectured projection inequality[J]. Appl. Math. Mech., 2007, 28(2):269-276.
[7] Wang Weidong, Leng Gangsong. On the monotonicity of Lp-centroid body[J]. J. Sys. Sci. Math.Scis., 2008, 28(2):154-162(in Chinese).
[8] Wang Weidong, Leng Gangsong. Some affine isoperimetric inequalities associated with Lp-affine surface area[J]. Houston J. Math., 2008, 34(2):443-453.
[9] Wang Weidong, Leng Gangsong. On the L_p-version of the Petty's conjectured projection inequality and applications[J]. Taiwan. J. Math., 2008, 12(5):1067-1086.
[10] Wang Weidong, Leng Gangsong. Inequalities of the quermassintegrals for the Lp-projection body and the Lp-centroid body[J]. Acta Math. Sci., 2010, 30B(1):359-368.
[11] Wang Weidong, Leng Gangsong, Lu Fenghong. On Brunn-Minkowski inequality for the quermassintegrals and dual quermassintegrals of Lp-projection bodies[J]. Chinese Math. Ann., 2008, 29A(2):209-220(in Chinese).
[12] Wang Weidong, Lu Fenghong, Leng Gangsong. A type of monotonicity on the Lp centroid body and Lp projection body[J]. Math. Inequal. Appl., 2005, 8(4):735-742.
[13] Wang Weidong, Lu Fenghong, Leng Gangsong. On monotonicity properties of the Lp-centroid bodies[J]. Math. Inequal. Appl., 2013, 16(3):645-655.
[14] Wang Weidong, Wei Daijun, Xiang Yu. On monotony for the L_p-projection body[J]. Chinese Adv.Math., 2008, 37(6):690-700(in Chinese).
[15] Wang Weidong, Wei Daijun, Xiang Yu. On reverses of the L_p-Petty projection inequality[J]. Chin.Quart. J. Math., 2009, 24(4):491-498.
[16] Ludwig M. Minkowski valuations[J]. Trans. Amer. Math. Soc., 2005, 357(10):4191-4213.
[17] Wang Weidong, Wan Xiaoyan. Shephard type problems for general Lp-projection bodies[J]. Taiwan.J. Math., 2012, 16(5):1749-1762.
[18] Wang Weidong, Feng Yibin. A general Lp-version of Petty's affine projection inequality[J]. Taiwan.J. Math., 2013, 17(2):517-528.
[19] Wang Weidong, Wang Jianye. Extremum of geometric functionals involving general Lp-projection bodies[J]. J. Inequal. Appl., 2016, 2016:1-16.
[20] Feng Yibin, Wang Weidong, Lu Fenghong. Some inequalities on general L_p-centroid bodies[J].Math. Inequal. Appl., 2015, 18(1):39-49.
[21] Wang Weidong, Leng Gangsong. The Petty projection inequality for L_p-mixed projection bodies[J]. Acta Math. Sinica(English Series), 2007, 23(8):1485-1494.
[22] Feng Yibin, Wang Weidong. The Shephard type problems and monotonicity for L_p-mixed centroid body[J]. Indian J. Pure Appl. Math., 2014, 45(3):265-283.
[23] Liu Lijuan, Wang Wei, He Binwu. Fourier transform and L_p-mixed projection bodies[J]. Bull.Korean Math. Soc., 2010, 47(5):1011-1023.
[24] Ma Tongyi. On Lp-mixed centroid bodies and dual L_p-mixed centroid bodies[J]. Acta Math. Sinica(Chinese Series), 2010, 53(2):301-314.
[25] Wang Weidong, Wan Xiaoyan. L_p-mixed projection bodies and Lp-mixed quermassintegrals[J]. J.Math. Inequal., 2014, 8(4):879-888.
[26] Wan Xiaoyan, Wang Weidong. Petty projection inequalities for the general L_p-mixed projection bodies[J]. Wuhan Univ. J. Nat. Sci.,2012,17(3):190-194.
[27] Firey W J. p-means of convex bodies[J]. Math. Scand., 1962, 10:17-24.
[28] Lutwak E. The Brunn-Minkowski-Firey theory II:affine and geominimal surface areas[J]. Adv.Math., 1996, 118(2):244-294.
[29] Firey W J. Mean cross-section measures of harmonic means of convex bodies[J]. Pacific J. Math.,1961, 11(4):1263-1266.
[30] Lutwak E. The Brunn-Minkowski-Firey theory I:mixed volumes and the Minkowski problem[J]. J.Diff. Geom., 1993, 38(1):131-150.
[31] Lutwak E. Dual mixed volumes[J]. Pacific J. Math., 1975, 58(2):531-538.
[32] Wang Weidong, Leng Gangsong. L_p-dual mixed quermassintegrals[J]. Indian J. Pure Appl. Math.,2005, 36(4):177-188.