平面上的逆Bonnesen-型Minkowski不等式
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摘要
研究了平面中形如A~2_(K,L)-A_KA_L≤U_(K,L)的Minkowski不等式的上界,即逆Bonnesen-型Minkowski不等式.设K,L是平面中的凸体,其面积分别为A_K,A_L,其中A_(K,L)为两凸体的混合面积,U_(K,L)为与K,L有关的几何不变量.利用平面上给定两凸体的支持函数,构造一类与给定凸体相关的新凸体.通过对新凸体几何性质的讨论,得到了一些新的加强的逆Bonnesen-型Minkowski不等式,并用此类不等式可推出一些已有结果.
We study in this paper the upper bound of the Minkowski inequality in the plane,i.e.a reverse Bonnesen-style Minkowski inequality,such as A_K~2,_L-A_KA_L≤U_K,_L.Let K and Lbe convex bodies whose areas are A_Kand A_L,respectively,and A_K,_Lis the mixes area of the two convex bodies and U_K,_Lis the geometric invariant related to K and L.We construct a class of convex body by the support function of the given convex bodies.By discussing the geometric properties of the new convex body,we obtain some new stronger reverse Bonnesen-style Minkowski inequalities and some results can be derived from those inequalities.
We study in this paper the upper bound of the Minkowski inequality in the plane,i.e.a reverse Bonnesen-style Minkowski inequality,such as A_K~2,_L-A_KA_L≤U_K,_L.Let K and Lbe convex bodies whose areas are A_Kand A_L,respectively,and A_K,_Lis the mixes area of the two convex bodies and U_K,_Lis the geometric invariant related to K and L.We construct a class of convex body by the support function of the given convex bodies.By discussing the geometric properties of the new convex body,we obtain some new stronger reverse Bonnesen-style Minkowski inequalities and some results can be derived from those inequalities.
引文
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[2] BURAGO Y D,ZALGALLER V A.Geometric Inequalities[M].Berlin:Springer-Verlag,1988.
[3] CROKE C B.A Sharp Four Dimensional Isoperimetric Inequality[J].Commentarii Mathematici Helvetici,1984,59(1):187-192.
[4] ENOMOTO K.A Generalization of the Isoperimetric Inequality on S2 and Flat Tori in S3[J].Proceedings of the American Mathematical Society,1994,120(2):553-558.
[5] GRINBER E L.Isoperimetric Inequalities and Identities for k-Dimensional Cross-Sections of Convex Bodies[J].Mathematische Annalen,1991,291(1):75-86.
[6]张增乐,罗淼,陈方维.平面上的新凸体与逆Bonnesen-型不等式[J].西南师范大学学报(自然科学版),2015,40(4):27-30.
[7] GYSIN L M.The Isoperimetric Inequality for Nonsimple Closed Curves[J].Proceedings of the American Mathematical Society,1993,118(1):197-203.
[8]杨林,罗淼,侯林波.逆的对偶Brunn-Minkowski不等式[J].西南大学学报(自然科学版),2016,38(4):85-89.
[9] HOWARD R.The Sharp Sobolev Inequality and the Banchoff-Pohl Inequality on Surfaces[J].Proceedings of the American Mathematical Society,1998,126(9):2779-2787.
[10]曾春娜,周家足,岳双珊.两平面凸域的对称混合等周不等式[J].数学学报(中文版),2012,55(2):355-362.
[11]王鹏富,徐文学,周家足,等.平面两凸域的Bonnesen型对称混合不等式[J].中国科学(数学),2015,45(3):245-254.
[12]LUO M,XU W X,ZHOU J Z.Translative Containment Measure and Symmetric Mixed Isohomothetic Inequalities[J].Science China Mathematics,2015,58(12):2593-2610.
[2] BURAGO Y D,ZALGALLER V A.Geometric Inequalities[M].Berlin:Springer-Verlag,1988.
[3] CROKE C B.A Sharp Four Dimensional Isoperimetric Inequality[J].Commentarii Mathematici Helvetici,1984,59(1):187-192.
[4] ENOMOTO K.A Generalization of the Isoperimetric Inequality on S2 and Flat Tori in S3[J].Proceedings of the American Mathematical Society,1994,120(2):553-558.
[5] GRINBER E L.Isoperimetric Inequalities and Identities for k-Dimensional Cross-Sections of Convex Bodies[J].Mathematische Annalen,1991,291(1):75-86.
[6]张增乐,罗淼,陈方维.平面上的新凸体与逆Bonnesen-型不等式[J].西南师范大学学报(自然科学版),2015,40(4):27-30.
[7] GYSIN L M.The Isoperimetric Inequality for Nonsimple Closed Curves[J].Proceedings of the American Mathematical Society,1993,118(1):197-203.
[8]杨林,罗淼,侯林波.逆的对偶Brunn-Minkowski不等式[J].西南大学学报(自然科学版),2016,38(4):85-89.
[9] HOWARD R.The Sharp Sobolev Inequality and the Banchoff-Pohl Inequality on Surfaces[J].Proceedings of the American Mathematical Society,1998,126(9):2779-2787.
[10]曾春娜,周家足,岳双珊.两平面凸域的对称混合等周不等式[J].数学学报(中文版),2012,55(2):355-362.
[11]王鹏富,徐文学,周家足,等.平面两凸域的Bonnesen型对称混合不等式[J].中国科学(数学),2015,45(3):245-254.
[12]LUO M,XU W X,ZHOU J Z.Translative Containment Measure and Symmetric Mixed Isohomothetic Inequalities[J].Science China Mathematics,2015,58(12):2593-2610.