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Inequalities for L_p Mixed Intersection Bodies_论文

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Chin.Quart.3。o/Math.  2018,33(2):156— 165  Inequalities for Lp M ixed Intersection B odies  LIU Li-juan,WANG Wei  School ol Mathematics and Computational Science.Hunan University o1 Science and Technology.  Xiangtan.411201 China;Department o{Mathematics Tongji University Shanghai 200092}China)  A bstract:Haberl and Ludwig defined the notions of Lp intersection bodies.In this paper,  we introduce the L口m ixed intersection bodies,and establish som e geometric inequa lities for  p m ixed intersection bodies.Furtherm ore,the Busem ann-Petty type problem f o r Lp m ixed  intersection bodies are shown.  K ey words:Star body;Radial function;Lp m ixed intersection bodies;LP Cosine tra n sform   2000 M R Subject Classif ication:52A40。52A20  CLC num ber:O 178  D ocum ent COde:A  Article ID:1002-0462(2018)02-0 156—10  §1.  Introduction  The unit ball and its surface in Euclidean n-spaceⅡ  are denoted by B and S 一 .respec—  tively.The radial function PK :S“  .-+[0,。。)of a compact star-shaped about the origin,  K ∈ Ⅱ ”,is def ined,for u ∈  一 ,by  p(K,U):pK(U)=max{A  0: Au∈  }.  If pK(·)is positive and continuous,then K is called a star body about the origin.The set  of star bodies about the origin in 酞  is denoted by  . The subset of  that contains the  origin-symmetric star bodies shall be denoted by霹 .Obviously,for K,L∈S“,  K  L{争PK(u)  pL(钆),V u∈S 一 .  (1.2)  is independent of u ∈ S 一 .then we say star bodies K  and L a re dilates  The notions of intersection bodies was introduced by Lutwak[211.The intersection body,  , of K  is the star body whose raidal function in the direction  ∈ S”一  is equal to the  R eceived date:2015—11-15  Foundation item :Supported by the Natural Science Foundation of Hunan Province(2017JJ3085+16C0635);  Supported by the China Postdoctoral Science Foundation(2016M601644)  Biographies:LIU Li-juan(1980-),female,Hailun,Heilongjiang,a lecturer of Hunan University of Science  and Technology,Ph.D.,engages in convex geometry;WANG Wei(1982一),male,Qiyang,Hunan,an associate  professor of H unan University of Science and Technology,Ph.D .,engages in convex geom etry.  No.2  LIU Li-juan et al:Inequalities for Lp M ixed Intersection Bodies  157  (n一1)一dimensional volume of the section of K by u上,the hyperplane orthogonal to乱,i.e.,for  all  ∈ S“一 ,  p(IK,钆)=vol(K n 上),  where vol denotes(n一1)一dimensional volume.  Intersection bodies have attracted increased interest during past two decades(see f2,5,7,11—  20,23,30f).  In particular,intersec



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