摘要
本学位论文致力于研究L_p-空间中凸体几何的度量不等式和极值问题,隶属于L_p-Brunn-Minkowski理论(又称为Brunn-Minkowski-Firey理论)领域,该领域是近十多年来在国际上发展非常迅速的一个几何学分支。本文主要利用L_p-Brunn-Minkowski理论的基本概念、基本知识和积分变换方法,研究了L_p-空间中凸体几何领域诸多几何体:L_2-投影体,L_p-截面体,混合截面体,混合新几何体Γ_(-p,i)K以及由我们新引入的L_p-仿射表面积、L_p-混合均质积分、L_p-混合仿射表面积所构成的度量不等式和极值问题。
利用∏_2K是一个中心在原点的椭球这一特殊性质,在第二章给出了投影不等式Petty猜想的L_p-形式(p=1时即是Petty投影不等式猜想本身)在当p=2时,其逆形式的两个结果,这些结果是对投影不等式Petty猜想的完善,同时建立了L_2-投影体∏_2K与标准化下经典投影体∏K的包含关系,还解答了一个约束最小化问题。
在第三章中,我们依据Gardner和Giannopoulos,V.Yaskin和M.Yaskin,C.Haberel和M.Ludwig提出的概念,合理地用不同的符号重新定义了L_p-截面体I_pK。对这一几何体,给出了算子I_p的线性等价性,及其当p≤-1时的单调性结果,同时将L_p-截面体和L_p-对偶混合均质积分(?)_(-p,i)(K,L)相结合,在正规化L_p-径向加或者L_p-径向线性组合的条件下,分别建立了关于L_p-截面体的对偶均质积分形式的Brunn-Minkowski不等式及其隔离加强形式。
Lutwak推广了Winterniz单调性问题,即得到:假设K∈F~n,且E是一个椭球,以及K的投影体的体积不超过E的投影体的体积,那么Ω(K)≤Ω(E)成立。在第四章,利用在L_p-混合体积和L_p-广义仿射表面积之间所建立的等价关系,我们把上述Lutwak的结果推广到L_p-形式,而且作为这种等价关系的应用,建立了L_p-混合均质积分形式的Aleksandrov的投影定理和Petty-Schneider定理。
在凸几何中,凸体的极体是一个非常重要的研究对象。在最重要的仿射等周不等式中,Blaschke-Santaló不等式和Petty投影不等式是两个与凸体的极体密切相关的不等式,可是对于星体而言,其极体却并不一定存在。在第五章,利用Moszy(?)ska介绍的星对偶概念,将混合截面体的星对偶与对偶均质积分结合起来,在调和p-组合或者p-径向线性组合的条件下,分别建立了关于混合截面体的星对偶的对偶均质积分形式的Brunn-Minkowski不等式。
根据Lutwak,Yang和Zhang提出的新几何体Γ_(-p)K,我们在第六章引入混合新几何体Γ_(-p,i)K的概念—新几何体是它的特殊情形。对混合新几何体Γ_(-p,i)K,我们获得了算子Γ_(-p,i)的五个基本性质,建立了Γ_(-p,i)K的体积V(Γ_(-p,i)K)与凸体K的体积V(K)之间的关系,以及关于混合新几何体Γ_(-p,i)K的Shephard类问题。
此外,我们还在第七章研究了L_p-混合仿射表面积和L_p-质心体的相互关系,获得了Busemann-Petty仿射质心不等式的广义L_p-形式,得到了关于L_p-混合仿射表面积的Blaschke-Santaló不等式,同时,获得了对偶Urysohn不等式的一般形式,最后,利用p-Blaschke平行体的定义,我们建立了与Marcus-Lopes,Bergstrom和Ky Fan不等式的一种类似形式。
The thesis is devoted to the study of metric inequalities and extremum problems in convex bodies geometry of L_p-space,and belongs to the domain,which is a high-speed developing geometry branch on the decade of late,of the L_p Brunn-Minkowski theory(or called Brunn-Minkowski-Firey theory).By applying the basic notions,basic theories and integral transforms of the L_p Brunn-Minkowski theory,we reseach the metric inequalities and extremum properties of some geometry bodies containing the L_2-projection body, L_p-intersection body,mixed intersection bodies,mixed new geometry bodyΓ_(p,i)K and a new notion from our definition—L_p—affine surface area,L_p-mixed quermassintegrals and L_p-mixed affine surface areas in the L_p-Brunn-Minkowski theory.
In Chapter two we use the special property thatΠ_2K is an origin-centered ellipsoid to give two versions for the reverses of the L_2-Petty projection inequality(L_1-Petty projection inequality is just the Petty's conjectured projection inequality),at the samr time, the inclusive relationships between L_2-projection bodyΠ_2K and the classical projection bodyΠK are established,a contrained minimization problem is solved.
In Chapter three,associated with the notions of Gardner's and Giannoponlos',or V.Yaskin's and M.yaskin's,or C.haberl's and M.Ludwig's,we reasonablly rewrite it usng some different signs—L_p-intersection body I_pK.On this geometry body,linearly equivalent property and several monotonicity results when p≤-1 are given,and together L_p-intersection body with Lp dual mixed quermassiintegrals(?)_(-p,i)(K,L),under the normalized L_p radial addition and L_p radial linear combination,we respectively show the dual quermassintegrals version's Brunn-Minkowski inequality and its isolate forms about L_p-intersection body.
Lutwak extended Winterniz monotonicity problem,that is,if K∈K~n and E is an ellipsoid,then if the areas of the projections of K do not exceed those of E,it follows thatΩ(K)≤Ω(E).Chapter four use an equivalent relationship between L_p-mixed volumes and L_p-extended affine surface areas,we extend Lutwak's result to L_p analog. As applications of this approach,we establish the L_p-mixed quermassintegrat version's Aleksandrov's projection theorem and Petty-Schneider theorem.
In the context of convex geometry,the polar of a convex body is an important object, however,the polar of a star body may not exist.In Chapter five,by the notion of star dual of a star body that Moszynska introduced,and associated with star dual of mixed intersection bodies and dual quermassintegrals,with the harmonic p-combination and p-radial linear combination,we respectively state the dual quermassintegrals version's Brunn-Minkowski inequality about star dual of mixed intersection bodies.
Recently,Lutwak,Yang and Zhang posed the notion of new geometric bodyΓ_(-p)K, in Chapter six,we introduce a new notion—the mixed new geometry bodyΓ_(-p,i)K(then new geometric body being its a special case).For this geometric body,we obtain five properties of the opertorΓ_(-p,i),establish the relationship between the volumes ofΓ_(-p,i)K and that of convex body K,and study Shephard version's problem about the mixed new geometric body.
As an aside,in Chapter seven,the concepts of the ith L_p-mixed anne surface area and L_p-polar curvature images are introduced,together L_p-centroid body with p-Blaschke linear combination,we prove the generalization of L_p-versions of the Busemann-Petty affine centroid inequality,and get the inequality analogous to the Blaschke-Santalóinequality for the L_p-mixed anne surface area and the general forms of the dual Urysohn inequality,and obtain the similar to the inequality for Marcus-Lopes,Bergstrom and Ky Fan.
引文
[1]Aleksandrov A.D.,On the theory of mixed volumes.Ⅰ.Extension of certain concepts in the theory of convex bodies[in Russian],Mat.Sbornik N.S.,1937,2:947-972.
[2]Aleksandrov A.D.,On the theory of mixed volumes.Ⅲ.Extension of two theorems of Minkowski on convex polyhedra to arbitrary convex bodies,Mat.Sbornik N.S.,1938,3:27-46.
[3]Aleksandrov A.D.,On the surface area measure of convex bodies,Mat.Sbornik N.S.,1939,6:167-174.
[4]Aleksandrov A.D.,Smoothness of the convex surface of bounded Gaussian curvature,C.R.(Dokl.) Acad.Sci.URSS,1942,36:195-199.
[5]Aubin T.,Problemes isoperimetriques et espaces de Sobolev,J.Diff.Geom.,1976,11:573-598.
[6]Aubin T.and Li Y.Y.,On the best Sobolev inequality,J.Math.Pure Appl.,1999,78:353-387.
[7]Ball K.,Volumes of sections of cubes and related problems,Israel Seminar(G.A.F.A.)(1988),Lecture Notes in Math.,Vol.1376,Springer-Verlag,Berlin and New York,(1989),251-260.
[8]Ball K.,Volume ratios and a reverse isoperimetric inequality,J.London Math.Soc.,1991,44:351-359.
[9]Bakry D.and Ledoux M.,Levy-Gromov's isoperimetric inequality for an infinite dimensional diffusion generator,Invent.Math.,1996,123:259-281.
[10]Barthe F.An extremal property of the mean width of the simplex,[J].Math.Ann.,1998,310(4):685-693.
[11]Beckner W.,Sharp Sobolev inequahties on the sphere and the Moser-Trudinger inequality,Annals Math.,1993,138:213-242.
[12]Beckner W.and Pearson M.,On sharp Sobolev embedding and the logarithmic Sobolev inequality,Bull.London Math.Soc.,1998,30:80-84.
[13]Blaschke W.,"Vorlesungen uber Differentialgeometrie.Ⅱ.Affine Differentialgeometrie,"Berlin:Springer,1923.
[14]Bobkov S.and Houdre C.,Some connections between isoperimetric and Sobolev-type inequalities,Memoirs of Amer.Math.Soc.,Num.1997,616.
[15]Bolker E.D.,A class of convex bodies,Trans.Amer.Math.Soc.,1969,145:323-345.
[16]Bonnesen,T.,Fenchel W.:Theory of convex bodies.BCS Associates.Moscow.ID.1987.German original:Springer.Berlin.1934.
[17]Bourgain J.and Milman V.D.,New volume ratio properties for convex symmetric bodies in R~n,Invent.Math.,1987,88:319-340.
[18]Bourgain J.and Lindenstrauss J.,Projection bodies,Geometric Aspects of Functional Analysis(Lindenstrauss J.and Milman V.D.,Eds)Springer Lecture Notes in Math.,1988,1317:250-270.
[19]Brannen N.S.Volumes of projection bodies[J].,Mathematka,1996,43(2):255-264.
[20]Brascamp H.J.,Lieb E.H.,On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems,including inequalities for log concave functions,and with an application to the diffusion equation,J.Functional Anal.1976,22:366-389.
[21]Brezis H.and Nirenberg L.,Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,Comm,Pure Appl.Math.,1983,36:437-477.
[22]Burago Y.D.and Zalgaller V.A.,Geometric Inequalities,Berlin:Springer-Verlag,1988.
[23]Busemann H.,A theorem on convx bodies of the Brunn-Minkowski type,Proc.Nat.Acad.Sci.U.S.A.,1949,35:27-31.
[24]Busemann H.,Volume in terms of concurrent cross-sections,Pacific J.Math.,1953,3:1-12.
[25]Caffarelli L.,Nirenberg L.and Spruck J.,The Drichlet problem for nonlinear second order elliptic equations I.Monge-Ampere equations,Comm.Pure Appl.Math.,1984,37:369-402.
[26]Calabi E.,Improper affine hyperspheres of convex type and a generalization of a theorem by K.Jorgens,Michigan Math.J.,1958,5:105-126.
[27]Campi S.and Grochi P.,The L~p-Busemann-Petty centroid inequality,Adv.Math.,2002,167:128-141.
[28]Campi S.and Grochi P.,On the reverse L~p-Busemann-Petty centroid inequality,Mathematika,2002,49:1-11.
[29]Carlen E.and Loss M.,Extremals of functions with competing symmetry,J.Funct.Anal.,1990,88:437-456.
[30]Carleson L.and Chang S.Y.A.,On the existence of an extremal function for an inequality of J.Moser,Bull.Sci.Math.,1986,110:113-127.
[31]Chakerian G.D.and Lutwak E.,Bodies with similar projections,Trans.Amer.Math.Soc.,1997,349:1811-1820.
[32]Cheng S.Y.and Yau S.T.,The n-dimensional Minkowski problem,Comm.Pure.Appl.Math.,1976,29:495-516.
[33]Chou K.S.and Wang X.J.,The L_p-Minkowski problem and the Minkowski problem in centroaffine geometry,Adv.Math.,2006,205(1):33-83.
[34]Druet O.,Optimal Sobolev inequalities of arbitrary order on Riemannian compact manifolds,J.Funct.Anal.,1998,159:217-242.
[35]Federer H.and Fleming W.,Normal and integral currents,Ann.Math.,1960,72:458-520.
[36]Federer H.,Geometric Measure Theory,Berlin:Springer-Verlag,1969.
[37]Fenchel W.and Jessen B.,Mengenfunktionen und konvexe Korper,Danske.Vid.Sel.Mat.-fys.Medd.,1938,16:1-31.
[38]Firey W.J.,Polar means of convex bodies and a dual to the Brunn-Minkowski theorem,Canad.J.Math.,1961,13:444-453.
[39]Firey W.J.,Mean cross-section measures of harmonic means of convex bodies,Pacific J.Math.,1961,11:1263-1266.
[40]Firey W.J.,p-means of convex bodies,Math Scand,1962,10:17-24.
[41]Gardner R.J.,Intersection bodies and the Busemann-Petty problem,Trans.Amer.Math.Soc.,1994,342:435-445.
[42]Gardner R.J.,A positive answer to the Busemann-Petty problem in three dimensions,Ann.Math.,1994,140:435-447.
[43]Gardner R.J.,Geometric Tomography,Cambridge:Cambridge Univ.Press,1995.
[44]Gardner R.J.and Giannopoulos A.A.,p-Cross-section bodies,Indiana Univ.Math.J.,1999,48:593-613.
[45]Gardner R.J.,Koldobsky A.and Schlumprecht T.,An analytic solution to the Busemann-Petty problem on sections of convex bodies,Ann.Math.,1999,149(2):691-703.
[46]Giannopoulos A.and Papadimitrakis M.,Isotropic surface area measures,Mathematika,1999,46:1-13.
[47]Gluck H.,The generalized Minkowski problem in differential geometry in the large,Ann.of Math.,1972,96:245-276.
[48]Gluck H.,Manifolds with preassigned curvature-a survey,Bull.Amer.Math.Soc.,1975,81:313-329.
[49]GoodeyP.R.and Well W.,Zonoids and generalizations,in "Handbook of Convex Geometry"(P.M.Gruber and J.M.Wills,Eds.),North-Holland,Amsterdam,1993.
[50]Grinberg E.and Zhang G.,Convolutions,transforms and convex bodies[J],Proc.London Math.Soc.,1999,78(3):77-115.
[51]Guan P.and Lin C.,On equation det(u_(ij)+δ_(iju))=u~pf on S~n,preprint,2000.
[52]Guleryuz O.G.,Lutwak E.,Yang D.and Zhang G.Y.,Information theoretic inequalities for contoured probability distributions,IEEE Trans.Inform.Theory,2002,48:2377-2383.
[53]Haberl C.and Ludwig M.,A characterization of L_p intersection bodies[J],Int.Math.Res.Not.,2006,Art.ID 10548,1-29.
[54]Hardy G.H.,Littlewood J.E.and Polya G.,Inequalities[M],Cambridge:Cambridge University Press,1959.
[55]Hebey E.and Vaugon M.,The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds,Duke Math.J.,1995,79:235-279.
[56]Hu C.,Ma X.N.and Shen C.,On the Christoffel-Minkowski problem of Firey's p-sum,Calc.Vat.Partial Differential Equations,2004,21(2):137-155.
[57]Hug D.,Lutwak E.,Yang D.and Zhang G.Y.,On the L_p Minkowski problem for polytopes,Discrete Comput.Geom.,2005,33:699-715.
[58]Koldobsky A.,Intersection bodies and the Busemann-Petty problem,C.R.Acad.Sci.Paris Ser.Ⅰ Math.,1997,325(11):1181-1186.
[59]Koldobsky A.,Intersection bodies,positive definite distributions,and the Busemann-Petty problem,Amer.J.Math.,1998,120(4):827-840.
[60]Leichtweiβ K.,Zur Affinoberflache konvexer korper,Manuscr Math.,1986,56:429-464.
[61]Leichtweiβ K.,Uber einige Eigenschaffen der Affinoberflache beliebiger Konvexer Korper,Results Math.,1988,13:255-282.
[62]Leichtweiβ K.,Bemerkungen zur Definition einer erweiterten Affinoberflache von E.Lutwak,Manuscripta Math.,1989,65:181-197.
[63]Leichtweiβ K.,On the history of the affine surface area for convex bodies,Results in Math.,1991,20:650-656.
[64]Leichtweiβ K.,Affine Geoetry of Convex Bodies,J.A.Barth,Heidelberg,1998.
[65]Leng G.S.and Zhang L.S.,Extreme properties of Quermassintegrals of convex bodies,Sci.in China(Ser.A),2001,44:837-845.
[66]Lewis D.R.,Finite dimensional subspaces of L_p,Studia Math.,1978,63(2):207-212.
[67]Lewis D.R.,Ellipsoids defined by Banaeh ideal norms,Mathematika,1979,26:18-29.
[68]Lewy H.,On the existence of a closed convex surface realizing a given Riemannian metric,Proc.Nat.Acad.Sci.USA.,1938,24:104-106.
[69]Lewy H.,On differential geometry in the large,I(Minkowski's problem),Trans.Amer.Math.Soc.,1938,43:258-170.
[70]Lieb E.H.,Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,Annals of Math.,1983,118:349-374.
[71]Lonke Y.,Derivatives of the L~p-cosine transform,Adv.Math.,2003,176(2):175-186.
[72]Ludwig M.,Projection bodies and valuations,Adv.Math.,2002,172:158-168.
[73]Ludwig M.,Valuations of polytopes containing the origin in their interiors,Adv.Math.,2002,170(2):239-256.
[74]Ludwig M.,Ellipsoids and matrix valued valuations,Duke Math.J.,2003,119:159-188.
[75]Ludwig M.,Minkowski valuations,Trans.Amer.Math.Soc.,2005,357:4191-4213.
[76]Lutwak E.,Selected affine isoperimetric inequalities,Handbook of convex geometry,North- Holland,Amsterdam,1996,pp.151-176.
[77]Lutwak E.,On the Blaschke-Santalo inequality,Ann.N.Y.Acad.Sci.,1985,440:106-112.
[78]Lutwak E.and Oliker V.,On the regularity of solutions to a generalization of the Minkowski problem,J.Differential Geom.,1995,41:227-246.
[79]Lutwak E.,Yang D.and Zhang G.Y.,The Cramer-Rao inequality for star bodies,Duke Math.J.,2002,112:59-81.
[80]Lutwak E.,Yang D.and Zhang G.Y.,A new ellipsoid associated with convex bodies,Duke Math.J.,2000,104:375-390.
[81]Lutwak E.,Yang D.and Zhang G.Y.,On L_p affine isoperimetric inequalities,J.Differential Geom.,2000,56:111-132.
[82]Lutwak E.,Yang D.and Zhang G.Y.,Sharp affine L_p Sobolev inequalities,J.Differential Geom.,2002,62(1):17-38.
[83]Lutwak E.,Yang D.and Zhang G.Y.,On the L_p-Minkowski problem,Trans.Amer.Math.Soc.,2004,356:4359-4370.
[84]Lutwak E.,Yang D.and Zhang G.Y.,L_p John ellipsoids,Proc.London Math.Soc.,2005,90:497-520.
[85]Lutwak E.,Yang D.and Zhang G.Y.,Volume inequalities for subspace of L_p,J.Differential Geom.,2004,68:159-184.
[86]Lutwak E.and Zhang G.Y.,Blaschke-Santalo inequalities,J.Differential Geom.,1997,47:,1-16.
[87]冷岗松,凸体的曲率映象与仿射表面积,数学学报,2002,45(4):797-802.
[88]Lutwak E.,Centroid bodies and dual mixed volumes,Proc.London Math.Soc.,1990,60:365-391.
[89]Lutwak E.,Mixed affine surface area,J.Math.Anal.Appl.,1987,125:351-360.
[90]Lutwak E.,Extended affine surface area,Adv.Math.,1991,85:39-68.
[91]Lutwak E.,On some affine isoperimetric inequalities,J.Differential Geom.,1986,23:1-13.
[92]Lutwak E.,Dual mixed volumes,Pacific J.Math.,1975,58:531-538.
[93]Lutwak E.,Intersection bodies and dual mixed volumes,Adv.Math.,1988,71:232-261.
[94]Lutwak E.,Inequalities for Hadwiger's harmonic Quermassintegrals,Math.Ann.,1988,28:165-175.
[95]Lutwak E.,The Brunn-Minkowski-Firey theory Ⅰ:mixed volumes and the Minkowski problem,J.Differential Geom.,1993,38:131-150.
[96]Lutwak E.,The Brunn-Minkowski-Firey theory Ⅱ:affine and geominimal surface areas,Adv.in Math.,1996,118:244-294.
[97]Lutwak E.,On a conjectured projection inequality of Petty[J],Contemp.Math.,1990,113:171-182.
[98]Maz'ya V.G.,Classes of domains and imbedding theorems for function spaces,Dokl.Akad.Nauk.SSSR,1960,133:527-530.
[99]Meyer M.and Werner E.,On the p-affine surface area,Adv.Math.,2000,152(2):288-313.
[100]Minkowski H.,Volumen und Oberfl'ache,Math.Ann.,1903,57:447-495[Also publish as Ges.Abh.Band 2(Teubner.Leipzig,(1911)) 230-276].
[101]Moszynska M.,Quotient star bodies,intersection bodies and star duality,J.Math.Anal.Appl.,1999,232:45-60.
[102]Neyman A.,Representation of L_p-norms and isometric embedding in L_p-spaces,Israel J.Math.,1984,48(2-3):129-138.
[103]Nirenberg L.,The Weyl and Minkowski problems in differential geometry in the large,Comm.Pure.Appl.Math.,1953,6:337-394.
[104]Osserman R.,The isoperimetric inequality,Bull.Amer.Math.Soc.,1978,84:1182-1238.
[105]Petty C.M.,Surface area of a convex body under atfine transformations,Proc.Amer.Math.Soc.,1961,12:824-828.
[106]Petty C.M.,Centroid surfaces,Pacific J.Math.,1961,11:1535-1547.
[107]Petty C.M.,Isoperimetric problems,In:Proc.Conf.Convexity and Combinatorial Geometry,Univ.Oklahoma,1972,26-41.
[108]Petty C.M.,Geominimal surface area,Geom.Dedicata,1974,3:77-97.
[109]Petty C.M.,Affine isoperimetric problems,In:Discrete geometry and convexity(eds J.E.Coodman,E.Lutwak,etc),Ann.New York Acad.Sci.,1985,440:113-127.
[110]Petty C.M.,Projection bodies,in:Proceedings,Coll Convexity,Copenhagen,(1965),234-241,Kφbenhavns Univ Mat Inst.(1967).
[111]Pisier G.,The volume of convex bodies and Banach space geometry,Cambridge:Cambridge Univ.Press,1989.
[112]Pogorelov A.V.,Regularity of a convex surface with given Gaussian curvature,Mat.Sb.,1952,31:88-103.
[113]Pogorelov A.V.,A regular solution of the n-dimensional Minkowski problem,Dokl.Akad.Nauk.SSSR 1971,199:785-788;English Transl.,Soviet Math.Dokl.,1971,12:1192-1196.
[114]Ryabogin D.and Zvavitch A.,The Fourier transform and Firey projections of convex bodies,Indiana Univ.Math.Journal,2004,53:667-682.
[115]Schneider R.,Convex Bodies:The Brunn-Minkowski Theory,Cambridge:Cambridge Univ.Press,1993.
[116]Schneider R.,Geometric inequalities for Poisson processes of convex bodies and cylinders[J],Results in Math.,1987,11(1-2):165-185.
[117]Schneider R.,Zu einem Problem von Shephard uber die Projectionen konvexer korper,Math.Z.,1967,101:71-82
[118]Schneider R.and Weil W.,Zonoids and related topics,in"Convexity and its Applications"(P.M.Gruber and J.M.Wills,Eds.),Birkhauser,Basel,1983,pp.296-317.
[119]Schoen R.and Yah S.T.,Lectures on Differential Geometry,Boston:International Press,1994.
[120]Schutt C.and Werner,E.,The convex floating body,Math.Scand,1990,66:275-290.
[121]Schutt C.and Werner,E.,Surface bodies and p-affine surface area,Adv.Math.,2004,187:,98-145.
[122]Schutt C.,On the affine surface area,Proc.Amer.Math.Soc.,1993,118:1213-1218.
[123]Singer D.,Preassigning curvature on the two-sphere,J.Diferential Geom.,1974,9:633-638.
[124]Stancu A.,The discrete planar L_0-Minkowski problem,Adv.Math.,2002,167(1):160-174.
[125]Stancu A.,On the number of solutions to the discrete two-dimensional L_0-Minkowski problem,Adv.Math.,2003,180(1):290-323.
[126]Thompson A.C.,Minkowski geometry,Cambridge:Cambridge Univ.Press,1996.
[127]Talenti G.,Best constant in Sobolev inequality,Ann.Mat.Pura.Appl.,1976,110;353-372.
[128]Umanskiy V.,On solvability of the two-dimensional L_p-Minkowski problem,Adv.Math.,2003,180(1):176-186.
[129]Wang W.and Leng G.,L_p-dual mixed quermassintegrals[J],Pure Appl.Math.,2005,36(4):177-188.
[130]Werner E.,Illumination bodies and affine surface area,Studia Math.,1994,110:257-269.
[131]Yaskin V.and Yaskina M.,Centeriod bodies and comparison of volume[J],Indiana Univ.Math.J.,2006,55(3):1175-1194.
[132]Yau S.T.,Sobolev inequality for measure space,Tsing Hua Lectures on geometry and analysis(Hsinchu,1990-1991),299-313,Internat.Press,Cambridge,MA,1997.
[133]Yu W.,Wu D.and Leng G.,Qusai Lp-intersection bodies[J],Acta Math.Sinica,2007,23(11):1937-1948.
[134]Yuan S.and Leng G.,Inequality for mixed intersection bodies[J],Chinese Ann.of Math.,Series B,2005,26(3):423-436..
[135]Zhang G.Y.,Restricted chord projection and affine inequalities,Geom.Dedicata,1991,39:213-222.
[136]Zhang G.Y.,Centered bodies and daul mixed volumes,Trans.Amer.Math.Soc.,1994,345:777-801.
[137]Zhang G.Y.,A positive answer to the Busemann-Petty problem in R~4,Ann.Math.,1999,149:535-543.
[138]Zhang G.Y.,The affine Sobolev inequality,J.Differential Geom.,1999,53:183-202.