Volume Difference Inequalities for the Polars of Mixed Complex Projection Bodies
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摘要
In this paper, based on the notion of mixed complex projection and generalized the recent works of other authors, we obtain some volume difference inequalities containing Brunn-Minkowski type inequality, Minkowski type inequality and Aleksandrov-Fenchel inequality for the polars of mixed complex projection bodies.
In this paper, based on the notion of mixed complex projection and generalized the recent works of other authors, we obtain some volume difference inequalities containing Brunn-Minkowski type inequality, Minkowski type inequality and Aleksandrov-Fenchel inequality for the polars of mixed complex projection bodies.
In this paper, based on the notion of mixed complex projection and generalized the recent works of other authors, we obtain some volume difference inequalities containing Brunn-Minkowski type inequality, Minkowski type inequality and Aleksandrov-Fenchel inequality for the polars of mixed complex projection bodies.
引文
[1]Minkowski H.Volume und Oberfl?che.Math.Ann.,1993,57:447-495.
[2]Firey W J.p-means of convex bodies.Math.Scand.,1962,10:17-24.
[3]Lutwak E.Centroid bodies and dual mixed volumes.Proc.London Math.Soc.,1990,60(2):365-391.
[4]Lutwak E.The Brunn-Minkowski-Firey theory I:mixed volumes and the minkowski problem.J.Differential Geom.,1993,38(1):131-150.
[5]Lutwak E.The Brunn-Minkowski-Firey theory II:affine and geominimal surface areas.Adv.Math.,1996,118(2):244-294.
[6]Lutwak E,Yang D,Zhang G Y.Orlicz projection bodies.Adv.Math.,2010,223(1):220-242.
[7]Lutwak E,Yang D,Zhang G Y.Orlicz centroid bodies.J.Differential Geom.,2010,84(2):365-387.
[8]Ludwig M.General affine surface areas.Adv.Math.,2010,224:2346-2360.
[9]Ludwig M.Minkowski valuaion.Trans.Amer.Math.Soc.,2005,357(10):4191-4213.
[10]Haberl C,Schuster F E.Asymmetric affine Lp-Sobolev inequalities.J.Funct.Anal.,2009,257:641-658.
[11]Haberl C,Schuster F E,Xiao J.An asymmetric affine P′olya-Szeg¨o principle.Math.Ann.,2012,352:517-542.
[12]Ludwig M,Reitzner M.A classification of SL(n)invariant valuations.Math.Ann.,2010,172:1219-1267.
[13]Zhu B C,Zhou J Z,Xu W X.Dual Orlicz-Brunn-Minkowski theory.Adv.Math.,2014,264:700-725.
[14]Gardner R J,Hug D,Weil W,Ye D P.The dual Orlicz-Brunn-Minkowski theory.J.Math.Anal.Appl.,2015,430(2):810-829.
[15]Abardia J,Bernig A.Projection bodies in complex vector spaces.Adv.Math.,2011,227:830-846.
[16]Gardner R J.Geometric Tomography.2nd edition.Cambridge:Cambridge Univ.Press,2006.
[17]Schneider R.Convex Bodies:the Brunn-Minkowski Theory.Cambridge:Cambridge University Press,1993.
[18]Bonnesen T,Fenchel W.Theoric der konvexen K¨orper.Berlin:Speinger-Verlag,1934.
[19]Petty C M.Projection bodies.Proc.Coll.Convexity,Copenhagen,1965,Copenhagen:Kobenhavns Univ.Math.Inst.,1967:234-241.
[20]Schneider R.Zu einem problem von shephard¨uber die projectionen konvexer k¨orper.Math.Z.,1967,101:71-82.
[21]Bolker E D.A class of convex bodies.Trans.Amer.Math.Soc.,1969,145:323-345.
[22]Abardia J.Different bodies in complex vector spaces.J.Funct.Anal.,2012,263(11):3588-3603.
[23]Huang Q Z,He B W,Wang G T.The busemann theorem for complex p-convex bodies.Arch.,2012,99(3):289-299.
[24]Koldobsky A,K¨onig H,Zymonopoulou M.The complex Busemann-Petty problem on sections of convex bodies.Adv.Math.,2008,218(2):352-367.
[25]Koldobsky A,Paouris G,Zymonopoulou M.Complex intersection bodies.J.Lond.Math.Soc.,2013,88(2):538-562.
[26]Koldobsky A,Zymonopoulou M.Extremal sections of complex lp-ball,0 [27]Rubin B.Comparison of volumes of convex bodies in real,complex,and quaternionic spaces.Adv.Math.,2010,225(3):1461-1498.
[28]Zymonopoulou M.The complex Busemann-Petty problem for arbiirary measures.Arch.Math.,2008,91(5):664-678.
[29]Wang W,He R G.Inequalities for mixed complex projection bodies Taiwan J.Math.,2013,17:1887-1899.
[30]Liu L J,Wang W,Huang Q Z.On polars of mixed complex projection bodies.Bull.Korean Math.Soc.,2015,52(2):453-465.
[31]Leng G S.The Brunn-Minkowski inequality for volume differences.Adv.Appl.Math.,2004,32:615-624.
[32]Zhao C J,Bencze M.The Aleksandrov-Fenchel type inequalities for volume differences.Balkan J.Geom.Appl.,2010,15:163-172.
[33]Lv S J.Dual Brunn-Minkowski inequality for volume differences.Geom.Dedicata,2010,145:169-180.
[34]Zhao C J.Volume differences of mixed complex projection bodies.Bull.Belgian Math.Soc.,2014,21(3):553-564.
[35]Bechenbach E F,Bellman R.Inequalities.2nd edition.Berlin:Springer-Verlag,1965.
[36]Zhao C J.On polars of Blaschke-Minkowski homomorphisms.Math.Scan.,2012,111(1):147-160.
[37]Hardy G H,Littlewood E F,P′olya G.Inequalities.Cambridge:Cambridge University Press,1934.
[2]Firey W J.p-means of convex bodies.Math.Scand.,1962,10:17-24.
[3]Lutwak E.Centroid bodies and dual mixed volumes.Proc.London Math.Soc.,1990,60(2):365-391.
[4]Lutwak E.The Brunn-Minkowski-Firey theory I:mixed volumes and the minkowski problem.J.Differential Geom.,1993,38(1):131-150.
[5]Lutwak E.The Brunn-Minkowski-Firey theory II:affine and geominimal surface areas.Adv.Math.,1996,118(2):244-294.
[6]Lutwak E,Yang D,Zhang G Y.Orlicz projection bodies.Adv.Math.,2010,223(1):220-242.
[7]Lutwak E,Yang D,Zhang G Y.Orlicz centroid bodies.J.Differential Geom.,2010,84(2):365-387.
[8]Ludwig M.General affine surface areas.Adv.Math.,2010,224:2346-2360.
[9]Ludwig M.Minkowski valuaion.Trans.Amer.Math.Soc.,2005,357(10):4191-4213.
[10]Haberl C,Schuster F E.Asymmetric affine Lp-Sobolev inequalities.J.Funct.Anal.,2009,257:641-658.
[11]Haberl C,Schuster F E,Xiao J.An asymmetric affine P′olya-Szeg¨o principle.Math.Ann.,2012,352:517-542.
[12]Ludwig M,Reitzner M.A classification of SL(n)invariant valuations.Math.Ann.,2010,172:1219-1267.
[13]Zhu B C,Zhou J Z,Xu W X.Dual Orlicz-Brunn-Minkowski theory.Adv.Math.,2014,264:700-725.
[14]Gardner R J,Hug D,Weil W,Ye D P.The dual Orlicz-Brunn-Minkowski theory.J.Math.Anal.Appl.,2015,430(2):810-829.
[15]Abardia J,Bernig A.Projection bodies in complex vector spaces.Adv.Math.,2011,227:830-846.
[16]Gardner R J.Geometric Tomography.2nd edition.Cambridge:Cambridge Univ.Press,2006.
[17]Schneider R.Convex Bodies:the Brunn-Minkowski Theory.Cambridge:Cambridge University Press,1993.
[18]Bonnesen T,Fenchel W.Theoric der konvexen K¨orper.Berlin:Speinger-Verlag,1934.
[19]Petty C M.Projection bodies.Proc.Coll.Convexity,Copenhagen,1965,Copenhagen:Kobenhavns Univ.Math.Inst.,1967:234-241.
[20]Schneider R.Zu einem problem von shephard¨uber die projectionen konvexer k¨orper.Math.Z.,1967,101:71-82.
[21]Bolker E D.A class of convex bodies.Trans.Amer.Math.Soc.,1969,145:323-345.
[22]Abardia J.Different bodies in complex vector spaces.J.Funct.Anal.,2012,263(11):3588-3603.
[23]Huang Q Z,He B W,Wang G T.The busemann theorem for complex p-convex bodies.Arch.,2012,99(3):289-299.
[24]Koldobsky A,K¨onig H,Zymonopoulou M.The complex Busemann-Petty problem on sections of convex bodies.Adv.Math.,2008,218(2):352-367.
[25]Koldobsky A,Paouris G,Zymonopoulou M.Complex intersection bodies.J.Lond.Math.Soc.,2013,88(2):538-562.
[26]Koldobsky A,Zymonopoulou M.Extremal sections of complex lp-ball,0
[28]Zymonopoulou M.The complex Busemann-Petty problem for arbiirary measures.Arch.Math.,2008,91(5):664-678.
[29]Wang W,He R G.Inequalities for mixed complex projection bodies Taiwan J.Math.,2013,17:1887-1899.
[30]Liu L J,Wang W,Huang Q Z.On polars of mixed complex projection bodies.Bull.Korean Math.Soc.,2015,52(2):453-465.
[31]Leng G S.The Brunn-Minkowski inequality for volume differences.Adv.Appl.Math.,2004,32:615-624.
[32]Zhao C J,Bencze M.The Aleksandrov-Fenchel type inequalities for volume differences.Balkan J.Geom.Appl.,2010,15:163-172.
[33]Lv S J.Dual Brunn-Minkowski inequality for volume differences.Geom.Dedicata,2010,145:169-180.
[34]Zhao C J.Volume differences of mixed complex projection bodies.Bull.Belgian Math.Soc.,2014,21(3):553-564.
[35]Bechenbach E F,Bellman R.Inequalities.2nd edition.Berlin:Springer-Verlag,1965.
[36]Zhao C J.On polars of Blaschke-Minkowski homomorphisms.Math.Scan.,2012,111(1):147-160.
[37]Hardy G H,Littlewood E F,P′olya G.Inequalities.Cambridge:Cambridge University Press,1934.