L_p-Minkowski问题椭球解的唯一性（英文）
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摘要
本文研究了L_p-Minkowski问题(解是中心在原点的椭球的假定下).利用支撑函数与高斯曲率的关系,获得了当p<1时椭球解的唯一性,推广了L_p-Minkowski问题以及L_p-和的Christoffel-Minkowski问题的唯一性结果.
In this paper, we study the Lp-Minkowski problem(under the assumption that the solutions are ellipsoids centered at the origin). Through the relation between support function and Gauss curvature, we obtain the uniqueness of ellipsoid solutions for p < 1, and generalize the uniqueness result for Lp-Minkowski problem and Christoffel-Minkowski problem of L_p-sum.
In this paper, we study the Lp-Minkowski problem(under the assumption that the solutions are ellipsoids centered at the origin). Through the relation between support function and Gauss curvature, we obtain the uniqueness of ellipsoid solutions for p < 1, and generalize the uniqueness result for Lp-Minkowski problem and Christoffel-Minkowski problem of L_p-sum.
引文
[1]Lutwak E.The Brunn-Minkowski-Firey theory I:mixed volumes and the Minkowski problem[J].J.Diff.Geom.,1993,38(1):131-150.
[2]Lutwak E,Oliker V.On the regularity of solutions to a generalization of the Minkowski problem[J].J.Diff.Geom.,1995,41(1):227-246.
[3]Guan Pengfei,Ma Xinan,Zhou Feng.The Christofel-Minkowski problem III:existence and convexity of admissible solutions[J].Commun.Pure Appl.Math.,2006,59(9):1352-1376.
[4]Stancu A.The discrete planar L_0-Minkowski problem[J].Adv.Math.,2002,167(1):160-174.
[5]Stancu A.On the number of solutions to the discrete two-dimensional L_0-Minkowski problem[J].Adv.Math.,2003,180(1):290-323.
[6]Lutwak E,Yang D,Zhang Gaoyong.On the L_p-Minkowski problem[J].Trans.Amer.Math.Soc.,2004,356(11):4359-4370.
[7]Hug D,Lutwak E,Yang D,Zhang Gaoyong.On the L_p-Minkowski problem for polytopes[J].Discr.Comput.Geom.,2005,33(4):699-715.
[8]Chou K S,Wang Xujia.The L_p-Minkowski problem and the Minkowski problem in centroaffine geometry[J].Adv.Math.,2006,205(1):33-83.
[9]Schneider R.Convex bodies:the Brunn-Minkowski theory[M].Cambridge:Cambridge University Press,2013.
[10]Boroczky K J,Lutwak E,Yang D,Zhang Gaoyong.The logarithmic Minkowski problem[J].J.Amer.Math.Soc.,2013,26(3):831-852.
[11]Zhu Guangxian.The logarithmic Minkowski problem for polytopes[J].Adv.Math.,2014,262:909-931.
[12]Jian Huaiyu,Lu Jian,Wang Xujia.Nonuniqueness of solutions to the Lp-Minkowski problem[J].Adv.Math.,2015,281:845-856.
[13]Huang Yong,Liu Jiakun,Xu Lu.On the uniqueness of L_p-Minkowski problems:the constant pcurvature case in R~3[J].Adv.Math.,2015,281:906-927.
[14]Guan Pengfei,Lin Changshou.On equation det(u_(ij)+δ_(ij)u)=u~pf on S~n[J].Manuscript,1999.
[15]Andrews B.Gauss curvature flow:the fate of the rolling stones[J].Invent.Math.,1999,138(1):151-161.
[16]B(o|¨)r(o|¨)czky K J,Hegedüs P,Zhu Guangxian.On the discrete logarithmic Minkowski problem[J].Int.Math.Res.Not.,2016,2016(6):1807-1838.
[17]B(o|¨)r(o|¨)czky K J,Lutwak E,Yang D,Zhang Gaoyong.The log-Brunn-Minkowski inequality[J].Adv.Math.,2012,231(3):1974-1997.
[18]Chen Wenxiong.L_p Minkowski problem with not necessarily positive data[J].Adv.Math.,2006,201(1):77-89.
[19]Haberl C,Lutwak E,Yang D,Zhang Gaoyong.The even Orlicz Minkowski problem[J].Adv.Math.,2010,224(6):2485-2510.
[20]Jian Huaiyu,Lu Jian,Zhu Guangxian.Mirror symmetric solutions to the centro-affine Minkowski problem[J].Calc.Var.Partial Differ.Equ.,2016,55(2):1-22.
[21]Zhu Guangxian.The centro-affine Minkowski problem for polytopes[J].J.Diff.Geom.,2015,101(1):159-174.
[22]Zhu Guangxian.The L_p Minkowski problem for polytopes for 0 [23]Zhu Guangxian.The L_p Minkowski problem for polytopes for p<0[J].Indiana Univ.Math.J.,arXiv:1602.07774..
[24]Zhu Guangxian.Continuity of the solution to the L_p Minkowski problem[J].Proc.Am.Math.Soc.,2017,145(1):379-386.
[25]Bonnesen T,Fenchel W.Theory of convex bodies[M].HK:BCS Associates,1986.
[26]Alexandrov A D.Zur theorie der gemischten volumina von konvexen korpern III[J].Mat.Sb.,1938,3(2746):2.
[27]Fenchel W,Jessen B.Mengenfunktionen und konvexe Korper[M].K(o|¨)benhavn:Levin og Munksgaard,1938.
[28]Lewy H.On differential geometry in the large,I(Minkowski's problem)[J].Trans.Amer.Math.Soc.,1938,43(2):258-270.
[29]Nirenberg L.The Weyl and Minkowski problems in differential geometry in the large[J].Commun.Pure Appl.Math.,1953,6(3):337-394.
[30]Pogorelov A V.On existence of a convex surface with a given sum of the principal radii of curvature[J].Uspekhi Matematicheskikh Nauk,1953,8(3):127-130.
[31]Calabi E.Improper affine hyperspheres of convex type and a generalization of a theorem by K.Jorgens[J].Michigan Math.J.,1958,5(2):105-126.
[32]Cheng S Y,Yau S T.On the regularity of the solution of the n-dimensional Minkowski problem[J].Commun.Pure Appl.Math.,1976,29(5):495-516.
[33]Pogorelov A V.The multidimensional Minkowski problem[M].Washington:Winston,1978.
[34]Caffarelli L A.Interior W~(2,p)-estimates for solutions of the Monge-Ampere equation[J].Ann.Math.,1990,131(1):135-150.
[35]Gardner R.The Brunn-Minkowski inequality[J].Bull.Amer.Math.Soc.,2002,39(3):355-405.
[36]Firey W J.p-means of convex bodies[J].Math.Scand.,1962,10:17-24.
[37]Firey W J.Shapes of worn stones[J].Math.,1974,21(1):1-11.
[38]Lu Jian,Wang Xujia.Rotationally symmetric solutions to the L_p-Minkowski problem[J].J.Diff.Equ.,2013,254(3):983-1005.
[39]Ivaki M N.A flow approach to the L-2 Minkowski problem[J].Adv.Appl.Math.,2013,50(3):445-464.
[40]Tzitzeica M G.Sur une nouvelle classe de surfaces[J].Rendiconti del Circolo Matematico di Palermo,1908,25(1):180-187.
[41]Leichtweiss K.On a problem of W.J.Firey in connection with the characterization of spheres[J].Math.Pannon.,1995,6(1):67-75.
[42]Petty C M.Affine isoperimetric problems[J].Ann.New York Acad.Sci.,1985,440(1):113-127.
[43]Chen Fangwei,Zhou Jiazu.Some inequalities on general dual mixed volumes[J].J.Math.,2010,30(3):473-479.
[44]Guan Pengfei,Ma Xinan,Trudinger N,Zhu Xiaohua.A form of Alexandrov-Fenchel inequality[J].Pure Appl.Math.Quart.,2010,4:999-1012.
[45]Hu Changqing,Ma Xinan,Shen Chunli,On the Christoffel-Minkowski problem of Firey's p-sum[J].Cal.Var.Part.Diff.Equ.,2004,21(2):137-155.
[46]Guan Pengfei,Ma Xinan.The Christoffel-Minkowski problem I:convexity of solutions of a Hessian equation[J].Invent.math.,2003,151(3):553-577.
[47]Urbas J I E.An expansion of convex hypersurfaces[J].J.Diff.Geom.,1991,33(1):91-125.
[2]Lutwak E,Oliker V.On the regularity of solutions to a generalization of the Minkowski problem[J].J.Diff.Geom.,1995,41(1):227-246.
[3]Guan Pengfei,Ma Xinan,Zhou Feng.The Christofel-Minkowski problem III:existence and convexity of admissible solutions[J].Commun.Pure Appl.Math.,2006,59(9):1352-1376.
[4]Stancu A.The discrete planar L_0-Minkowski problem[J].Adv.Math.,2002,167(1):160-174.
[5]Stancu A.On the number of solutions to the discrete two-dimensional L_0-Minkowski problem[J].Adv.Math.,2003,180(1):290-323.
[6]Lutwak E,Yang D,Zhang Gaoyong.On the L_p-Minkowski problem[J].Trans.Amer.Math.Soc.,2004,356(11):4359-4370.
[7]Hug D,Lutwak E,Yang D,Zhang Gaoyong.On the L_p-Minkowski problem for polytopes[J].Discr.Comput.Geom.,2005,33(4):699-715.
[8]Chou K S,Wang Xujia.The L_p-Minkowski problem and the Minkowski problem in centroaffine geometry[J].Adv.Math.,2006,205(1):33-83.
[9]Schneider R.Convex bodies:the Brunn-Minkowski theory[M].Cambridge:Cambridge University Press,2013.
[10]Boroczky K J,Lutwak E,Yang D,Zhang Gaoyong.The logarithmic Minkowski problem[J].J.Amer.Math.Soc.,2013,26(3):831-852.
[11]Zhu Guangxian.The logarithmic Minkowski problem for polytopes[J].Adv.Math.,2014,262:909-931.
[12]Jian Huaiyu,Lu Jian,Wang Xujia.Nonuniqueness of solutions to the Lp-Minkowski problem[J].Adv.Math.,2015,281:845-856.
[13]Huang Yong,Liu Jiakun,Xu Lu.On the uniqueness of L_p-Minkowski problems:the constant pcurvature case in R~3[J].Adv.Math.,2015,281:906-927.
[14]Guan Pengfei,Lin Changshou.On equation det(u_(ij)+δ_(ij)u)=u~pf on S~n[J].Manuscript,1999.
[15]Andrews B.Gauss curvature flow:the fate of the rolling stones[J].Invent.Math.,1999,138(1):151-161.
[16]B(o|¨)r(o|¨)czky K J,Hegedüs P,Zhu Guangxian.On the discrete logarithmic Minkowski problem[J].Int.Math.Res.Not.,2016,2016(6):1807-1838.
[17]B(o|¨)r(o|¨)czky K J,Lutwak E,Yang D,Zhang Gaoyong.The log-Brunn-Minkowski inequality[J].Adv.Math.,2012,231(3):1974-1997.
[18]Chen Wenxiong.L_p Minkowski problem with not necessarily positive data[J].Adv.Math.,2006,201(1):77-89.
[19]Haberl C,Lutwak E,Yang D,Zhang Gaoyong.The even Orlicz Minkowski problem[J].Adv.Math.,2010,224(6):2485-2510.
[20]Jian Huaiyu,Lu Jian,Zhu Guangxian.Mirror symmetric solutions to the centro-affine Minkowski problem[J].Calc.Var.Partial Differ.Equ.,2016,55(2):1-22.
[21]Zhu Guangxian.The centro-affine Minkowski problem for polytopes[J].J.Diff.Geom.,2015,101(1):159-174.
[22]Zhu Guangxian.The L_p Minkowski problem for polytopes for 0
[24]Zhu Guangxian.Continuity of the solution to the L_p Minkowski problem[J].Proc.Am.Math.Soc.,2017,145(1):379-386.
[25]Bonnesen T,Fenchel W.Theory of convex bodies[M].HK:BCS Associates,1986.
[26]Alexandrov A D.Zur theorie der gemischten volumina von konvexen korpern III[J].Mat.Sb.,1938,3(2746):2.
[27]Fenchel W,Jessen B.Mengenfunktionen und konvexe Korper[M].K(o|¨)benhavn:Levin og Munksgaard,1938.
[28]Lewy H.On differential geometry in the large,I(Minkowski's problem)[J].Trans.Amer.Math.Soc.,1938,43(2):258-270.
[29]Nirenberg L.The Weyl and Minkowski problems in differential geometry in the large[J].Commun.Pure Appl.Math.,1953,6(3):337-394.
[30]Pogorelov A V.On existence of a convex surface with a given sum of the principal radii of curvature[J].Uspekhi Matematicheskikh Nauk,1953,8(3):127-130.
[31]Calabi E.Improper affine hyperspheres of convex type and a generalization of a theorem by K.Jorgens[J].Michigan Math.J.,1958,5(2):105-126.
[32]Cheng S Y,Yau S T.On the regularity of the solution of the n-dimensional Minkowski problem[J].Commun.Pure Appl.Math.,1976,29(5):495-516.
[33]Pogorelov A V.The multidimensional Minkowski problem[M].Washington:Winston,1978.
[34]Caffarelli L A.Interior W~(2,p)-estimates for solutions of the Monge-Ampere equation[J].Ann.Math.,1990,131(1):135-150.
[35]Gardner R.The Brunn-Minkowski inequality[J].Bull.Amer.Math.Soc.,2002,39(3):355-405.
[36]Firey W J.p-means of convex bodies[J].Math.Scand.,1962,10:17-24.
[37]Firey W J.Shapes of worn stones[J].Math.,1974,21(1):1-11.
[38]Lu Jian,Wang Xujia.Rotationally symmetric solutions to the L_p-Minkowski problem[J].J.Diff.Equ.,2013,254(3):983-1005.
[39]Ivaki M N.A flow approach to the L-2 Minkowski problem[J].Adv.Appl.Math.,2013,50(3):445-464.
[40]Tzitzeica M G.Sur une nouvelle classe de surfaces[J].Rendiconti del Circolo Matematico di Palermo,1908,25(1):180-187.
[41]Leichtweiss K.On a problem of W.J.Firey in connection with the characterization of spheres[J].Math.Pannon.,1995,6(1):67-75.
[42]Petty C M.Affine isoperimetric problems[J].Ann.New York Acad.Sci.,1985,440(1):113-127.
[43]Chen Fangwei,Zhou Jiazu.Some inequalities on general dual mixed volumes[J].J.Math.,2010,30(3):473-479.
[44]Guan Pengfei,Ma Xinan,Trudinger N,Zhu Xiaohua.A form of Alexandrov-Fenchel inequality[J].Pure Appl.Math.Quart.,2010,4:999-1012.
[45]Hu Changqing,Ma Xinan,Shen Chunli,On the Christoffel-Minkowski problem of Firey's p-sum[J].Cal.Var.Part.Diff.Equ.,2004,21(2):137-155.
[46]Guan Pengfei,Ma Xinan.The Christoffel-Minkowski problem I:convexity of solutions of a Hessian equation[J].Invent.math.,2003,151(3):553-577.
[47]Urbas J I E.An expansion of convex hypersurfaces[J].J.Diff.Geom.,1991,33(1):91-125.