ORLICZ MIXED INTERSECTION BODIES

Transcription

1 Vol. 37 ( 07 No. 3 J. of Math. (PRC ORICZ MIXED INTERSECTION BODIES GUO Yuan-yuan, MA Tong-yi, GAO i (.School of Mathematics and Statistics, Northwest Normal University, anzhou , China (.School of Mathematics and Statistics, Hexi University, Zhangye , China Abstract: In this paper, we investigate the Orlicz mixed intersection bodies and its properties. By using geometric analysis, we introduce the concept of Orlicz mixed intersection bodies and obtain the continuity and affine invariant property of the Orlicz mixed intersection bodies operator. By applying the integral methods and Steiner symmetrization, the affine isoperimetric inequality for Orlicz mixed intersection bodies is established. eywords: convex bodies; Orlicz mixed intersection bodies; Steiner symmetrization; affine isoperimetric inequality 00 MR Subject Classification: 5A0; 5A40 Document code: A Article ID: ( Introduction et n denote the set of convex bodies (compact, convex subsets with nonempty interiors in Euclidean n-space R n ; et o n denote the set of convex bodies containing the origin in their interiors in R n ; So n denotes the set of star bodies (about the origin in R n. We use to denote the n-dimensional volume of a body. Besides, write S n for the unit sphere in R n. For a compact set R n which is star-shaped with respect to the origin, we will use ρ ρ(, : R n \{o} [0, to denote its radial function. That is, ρ (x ρ(, x max{λ 0 : λx } for all x R n \{o}. If ρ is continuous and positive, then is called star body. In [], utwak introduced the intersection body I of each S n o, which is the star body with radial function ρ(i, u v( u, u S n, where v( denotes (n -dimensional volume and u is the hyperplane orthogonal to u. Received date: Accepted date: Foundation item: Supported by the National Natural Science Foundation of China (5600; the Science and Technology Plan of the Gansu Province (45RJZG7. Biography: Guo Yuanyuan (990, female, born at Shenqiu, Henan, postgraduate, major in convex geometric analysis and geometric inequality theory. Corresponding author: Ma Tongyi.

2 568 Journal of Mathematics Vol. 37 For 0 < p <, the p -intersection body I p of So n is defined by (see, e.g., Cardner and Giannopoulos [], Yuan [3] ρ(i p, u p x u p dx, u S n, (. where x u is the usual inner product of x, u R n, and integration is with respect to ebesgue measure. Haberl and udwig [4] also gave the following definition of p -intersection body for convex polytopes and investigated its characterization. There is a different constant between the p -intersection body of (. and the following p -intersection body in [5]. For u S n and 0 < p <, the p -intersection body of So n is defined by ρ(i p, u p Γ( p x u p dx, where Γ denotes the Gamma function. Intersection body [] played an important role for the solution of the celebrated Busemann- Petty problem, and found applications in geometric tomography [6], affine isoperimetric inequalities [7, 8] and the geometry of Banach spaces [9]. For more details about intersection body we refer the reader to [0 4]. Progress towards an Orlicz Brunn-Minkowski theory was made by utwak, Yang and Zhang [5, 6]. This theory plays such a crucial role that it is undeniably applied to a number of areas of geometry. The more development of the Orlicz Brunn-Minkowski theory can be found in [7 3]. Recently, Ma et al. [3] considered convex function : R\{o} (0, such that lim (t 0, lim (t and (0. This means that must be increasing t t 0 on (, 0 and decreasing on (0,. They assumed that is either strictly increasing on (, 0 or strictly decreasing on (0,, and denoted by Φ the class of such. Further, for So n, they gave the concept of Orlicz intersection body I as the star body whose radial function is given by ρ I (x sup {λ > 0 : y λ } dy, x R n, (. when (t t p, 0 < p <, then I I p. In this paper, we also consider the above convex function and assume that, are two star bodies (about the origin in R n with volume and. If Φ, then the Orlicz mixed intersection bodies I (, of, as the star body whose radial function at x R n is given by ρ I (, (x sup {λ > 0 : ( } x (y z λ, (.3 when (t t p with 0 < p <, and in (.3, then I (, I p (,, where the radial function of I p (, is given by ρ(i p (,, x p x (y z p.

3 No. 3 Orlicz mixed intersection bodies 569 Our main result deals with the affine isoperimetric inequality for Orlicz mixed intersection bodies. Here and in the following, for n, we denote by B the n-ball with the same volume as centered at the origin. Theorem. If, o n and Φ, then I (, I (B, B with equality if and are dilates of each other and have the same midpoints. Notations and Orlicz Mixed Intersection Bodies For a body n o, its support function, h h(, : S n R is defined by h (u max{u x : x }, u S n. For c > 0, the support function of the convex body c {cy : y } is h c (x ch (x, x R n. et G(n denote the group of linear transformations. For T G(n, write T t for the transpose of T, T for the inverse of T and T t for the inverse of the transpose of T. The support function of the image T {T y : y } is given by h T (x h (T t x, x R n. According to the definition of radial function ρ, for S n o and a > 0, we easily get ρ (ax a ρ (x and ρ a (x aρ (x. (. For T G(n, the radial function of the image T {T y : y } of S n o is given by ρ(t, x ρ(, T x for all x R n. (. The radial distance between, S n o is defined by δ(, sup ρ(, u ρ(, u. u S n For S n o, define the real number R and r by R max x ρ (x, r min x ρ (x. (.3 For n o, then the polar body of is defined by {x R n : x y, y }. It is easily proved that ( if n o.

4 570 Journal of Mathematics Vol. 37 From the definitions of support and radial function, it follows obviously that for each n o, we easily get h (u ρ (u and ρ (u h (u for all u S n. In the following write x, y for vectors in R n and x, y for vectors in R n. We will use (x, s, (y, s for vectors in R n R n R. For a convex body and a direction u S n, let u denote the image of the orthogonal projection of onto u, the subspace of R n orthogonal to u. The undergraph and overgraph functions, l u (, : u R and lu (, : u R, of in the direction u are given by {y + tu : l u (, y t l u (, y for y u }. Therefore the Steiner symmetral S u of o n in the direction u is defined by the body whose orthogonal projection onto u is identical to that of and whose undergraph and overgraph functions are l u (S u, y l u (S u, y l u (, y + l u (, y. (.4 For more details about the Steiner symmetrization we can refer to [33]. The following lemma will be required. emma. (see [6], emma. Suppose n o and u S n. For y relint u, the overgraph and undergraph functions of in direction u are given by and lu (, y min x u {h (x, x y } l u (, y min x u {h (x, x y }. From the definition of Orlicz mixed intersection bodies (.3 and is strictly increasing on (, 0 or strictly decreasing on (0,, we have the function ( x (y z λ λ is strictly increasing on (0, and it is also continuous. Thus we have emma. If, So n, then for u 0 S n, ( u0 (y z if and only if ρ I (u (, 0. emma.3 (see [34] et x (x, x,, x n R n and y (y, y,, y n R n. If 0 < m x k M, 0 < m y k M, k,, n, then ( n k x k ( n k y k ( M M m m + m m ( n M M x k y k. k

5 No. 3 Orlicz mixed intersection bodies 57 emma.3 implies that if x, y, z R n, then there exists a constant c 0 (0, such that x (y z c 0 x y z. (.5 If and are bodies in R n, their multiplicative d ab (, is defined by (see [35] d ab (, inf{ab : a, b > 0, b, a}. Whenever we write a x x b x, b y y a y. (.6 Denote by c > 0 the constant to meet c min{c > 0 : max{(c, ( c} } for each Φ. And denote distant d(, of convex bodies, by d(, : max{ x y : x, y }. (.7 emma.4 If, S n o and Φ, then for all u S n, there exists a positive constant b and c 0 (0, such that c 0 c b br ρ r I (, (u d(, c. Proof et x 0 S n and ρ I (, (x 0. Then 0 (y z. (.8 We first obtain the upper estimate. From the definition of c, either (c or ( c. If (c, by (.7, we have (c ( x0 (y z ( d(, ( d(,. With the definition of, is strictly increasing on (, 0 or strictly decreasing on (0,. So we can immediately obtain c d(,, i.e., d(, c. On the other hand, we obtain the lower estimate. Note that (c, x 0 S n, from

6 57 Journal of Mathematics Vol. 37 (.5, (.6 and (.3, it follows that (c ( x0 (y z ( c0 y z ( c0 y z ( c0 y b b z ( c0 bρ (y b ρ (z ( c0 b br r ( c0 b br, r where b > 0 and c 0 (0,. Since is strictly decreasing on (0,, we immediately get c c0 br b, r i.e., λ0 c0 c br b. r We complete the proof. The following shows that the Orlicz mixed intersection bodies operator I : So n So n is continuous. emma.5 Suppose Φ. If i, i So n and i So n, i So n, then I ( i, i I (,. Proof For Φ and is convex, continuous, and either strictly increasing on (, 0 or strictly decreasing on (0,, then for u 0 S n, we will show that ρ I ( i, i (u 0 ρ I (, (u 0. Suppose ρ I ( i, i (u 0 λ i, by emma., we have i i i i ( u0 (y z λ i. (.9 From emma.4, there exists a positive constant b and c 0 (0, such that c 0 b br i λ i d( i, i. c c r i Since i So n, i So n, we have d( i, i d(, > 0, R i R > 0, r i r > 0 and there exist α, β such that 0 < α λ i β < for all i. et {λ } be a convergent subsequence of {λ i }, and suppose that λ, together with the continuous of and (.9, we have ( u0 (y z,

7 No. 3 Orlicz mixed intersection bodies 573 which by emma. yields ρ I (, (u 0. This shows that ρ I ( i, i (u 0 ρ I (, (u 0. But for radial function on S n pointwise and uniform convergence are equivalent (see Schneider [8], p.54. Thus the pointwise convergence ρ I ( i, i ρ I on (, Sn completes the proof.. We next show that the Orlicz mixed intersection bodies operator is also continuous in emma.6 For, S n o and i Φ, then I i (, I (,. Proof et, S n o and u 0 S n. We will show that ρ I i (, ρ I (,. For Φ with is convex, continuous, and either strictly increasing on (, 0 or strictly decreasing on (0,, and let ρ I i (, (u 0 λ i, i.e., ( u0 (y z i, (.0 λ i then together with emma.4 we have that there exists a positive constant b and c 0 (0, such that c 0 c i b br λ i r d(, c i. Since i Φ, we have c i c > 0 and thus there exist α, β such that 0 < α λ i β < for all i. Denote by {λ } the arbitrary convergent subsequence of {λ i }, and suppose that λ, together with (.0, we immediately have ( u0 (y z, which by emma. yields ρ I (, (u 0. This shows that ρ I i (, (u 0 ρ I (, (u 0. Since the radial function ρ Ii (, ρ I(,(u 0 pointwise on S n they converge uniformly and hence I i (, I (,. emma.7 Suppose Φ. For, So n and a linear transformation T G(n, then I (T, T T t (I (,. Proof Suppose x 0 R n and ρ (I (T, T, x 0. (. et s T y, t T z, then T dett, T dett, ds dett dy, dt dett dz,

8 574 Journal of Mathematics Vol. 37 where dett is the absolute value of the determinant of T. From emma., we have T T ( x0 (s t dsdt T ( x0 (T y T z dett ( (x0 T y (x 0 T z ( (T t x 0 y (T t x 0 z ( T t x 0 (y z, T dett i.e., ρ I (, (T t x 0. By (., we have ρ I (, (T t x 0 ρ T t (I (, (x 0. Combining with equality (., we immediately obtain ρ (I (T, T, x 0 ρ (T t (I (,, x 0. That is, I (T, T T t (I (,. 3 Proofs of Main Results emma 3. Suppose Φ is strictly convex and, o n. If u S n and x, x u, then and I (S u,s u (x + x, ρ I (, (x, + ρ I (, (x, (3. ρ ρ I (S u,s u (x + x, ρ I (, (x, + ρ I (, (x,. (3. Equality in either inequality holds if ρ I(,(x, ρ I(,(x, with and are dilates having the same midpoints. Proof We only prove (3.. Inequality (3. can be established in the same way. For each z u, y u, let σ z σ z (u (z + Ru and σ y σ y (u (y + Ru be the lengths of the chords (z + Ru, (y + Ru. Define m z m z (u by m z (u l u (, z + l u (, z such that z + m z u is the midpoint of the chord (z + Ru. And define m y m y (u by m y (u l u (, y + l u (, y such that y + m y u is the midpoint of the chord (y + Ru. Note that the midpoints of the chords of in the direction u lie in a subspace if and only if there exists a µ u such that µ z m z for all z u. Similarly, the midpoints of the chords of in the direction u lie in a subspace if and only if there exists a µ u such that

9 No. 3 Orlicz mixed intersection bodies 575 µ y m y for all y u. If λ > 0, then we have ( (x, (y z λ ( (x, ((y, s (z, t d(y, sd(z, t u u S u mz + σ z m z σ z σ z σ z S u u u σ y σ y λ my + σ y m y σ y (y z + (s t λ (y z + m y + s m z t λ (y z + (s t + (m y m z by making the change of variables s m y + s and t m z + t. On the other hand, for λ > 0, we have u u S u ( ( x, (y z λ ( ( x, ((y, s (z, t d(y, sd(z, t mz + σ z m z σ z σ z σ z S u u u σ y σ y λ my + σ y m y σ y λ (y z (s t λ (y z m y + s + m z t λ (y z + (s t (m y m z by making the change of variables s m y s and t m z t. et λ x 0 x + x and λ + λ. dy dsdz dt dy ds dz dt d(y, s d(z, t (3.3 dy dsdz dt dy ds dz dt d(y, s d(z, t (3.4 From the convexity of, it follows that ( x 0 (y z + (s t λ ( x (y z + (s t + (m y m z + λ λ (y z + (s t (m y m z λ. (3.5

10 576 Journal of Mathematics Vol. 37 By (3.3, (3.4 and (3.5, it yields that ( λ (x, (y z + λ ( (x, (y z λ λ λ ( x (y z + (s t + (m y m z S u S u λ + λ ( x (y z + (s t (m y m z S u S u λ ( x 0 (y z + (s t d(y, s d(z, t S u S u ( (x 0, (y z et S u S u From emma., we get and d(y, s d(z, t d(y, s d(z, t. (3.6 ρ I (, (x, λ and ρ I (, (x, λ. ( (x, (y z λ ( (x, (y z λ. Combining with the fact that S u, S u and (3.6, we get S u S u S u S u In light of the continuity of and (.3 yields ( (x 0, (y z. ρ I (S u, S u ( x + x, λ + λ with the equality requiring equality in (3.5 for all z u, y u, and s [ σ y, σ y ], t [ σ z, σ z ]. Since is strictly convex, this means that we must have can not be linear in a neighborhood of the origin given by x (y z + (s t + (m y m z λ x (y z + (s t (m y m z λ (3.7 for all s ( σ y, σ y, t ( σ z, σ z. Choosing λ ρ I (, (x, and λ ρ I (, (x,, equation (3.7 immediately yields ρ I (, (x, λ λ ρ I (, (x,

11 No. 3 Orlicz mixed intersection bodies 577 and (x x y m y, (x x z m z for all y u and z u. But this means that the midpoints {(y, m y : y u } and {(z, m z : z u } of the chords of, parallel to u lie in the subspaces {(y, x x y : y u } and {(z, x x z : z u } of R n, respectively. As we can observe the equality holds if ρ I(,(x, ρ I(,(x, with and are dilates having the same midpoints. emma 3. Suppose, n o, Φ. If u S n, then I (S u, S u S u (I (,. Proof et y relint(i (, u. According to emma., there exist x x (y and x x (y in u such that lu (I (,, y h I (,(x, x y (3.8 and l u (I (,, y h I (,(x, x y. (3.9 From (.4, (3.8, (3.9 and emma 3., it follows that lu (S u I (,, y l u (I (,, y + l u (I (,, y (h I (, (x, x y + (h I (, (x, x y h I (, (x, + h I (, (x, ( x + x y h I (S u,s u( x + x, ( x + x y min {h I x u (S u,s u(x, x y } l u (I (S u, S u, y (3.0 and l u (S u I (,, y l u (I (,, y + l u (I (,, y (h I (, (x, x y + (h I (, (x, x y h I (, (x, + h I (, (x, ( x + x y h I (S u,s u( x + x, ( x + x y min {h I x u (S u,s u(x, x y } l u (I (S u, S u, y, (3.

12 578 Journal of Mathematics Vol. 37 (3.0 and (3. give the inclusion. Proof of Theorem. Combining with the Steiner symmetrization argument, there is a sequence of direction {u i }, such that the sequences { i } and { i } converge to B and B, respectively, where the sequences { i } and { i } are defined by i S ui S u and i S ui S u with i and i. Thus B and B. Since the Steiner symmetrization keeps the volume, by emma 3., we have I ( i, i I (S ui i, S ui i I ( i, i I (,, when i, we have I (B, B I (,. According to the equality condition of emma 3., above equality holds if and are dilates of each other and have the same midpoints. References [] utwak E. Intersection bodies and dual mixed volumes[j]. Adv. Math., 988, 7(: 3 6. [] Gardner R J, Giannopoulos A A. p-cross-section bodies[j]. Indiana Univ. Math. J., 999, 48(: [3] Yuan Jun, Cheung W. p-intersection bodies[j]. J. Math. Anal. Appl., 008, 338(: [4] Haberl C, udwig M. A characterization of p -intersection bodies[j]. Int. Math. Res. Not., 006, 006: 30. [5] Haberl C. p intersection bodies[j]. Adv. Math., 008, 7(6: [6] Gardner R J. Geometric tomography, second edition[m]. New York: Cambridge Univ. Press, 006. [7] oldobsky A, Paouris G, Zymonopoulou M. Isomorphic properties of intersection bodies[j]. J. Funct. Anal., 0, 6(9: [8] Schneider R. Convex bodies: the Brunn-Minkowski theory (nd ed.[m]. Cambridge: Cambridge Univ. Press, 04. [9] Thompson A C. Minkowski geometry[m]. Cambridge: Gambridge Univ. Press, 996. [0] udwig M. Intersection bodies and valuations[j]. Amer. J. Math., 006, 8: [] Gardner R J. Intersection bodies and the Busemann-Petty problem[j]. Trans. Amer. Math. Soc., 994, 34(: [] Gruber P M, udwig M. A Helmholtz-ie type characterization of ellipsoids II[J]. Disc. Comput. Geom., 996, 6(: [3] Wu Denghui, Ma Tongyi, Bu Zhenhui. The analog of p-busemann-petty inequality of p-mixed intersection body[j]. J. Math., 03, 33(6: [4] Zhao Changjian, eng Gangsong. Brunn-Minkowski inequality for mixed intersection bodies[j]. J. Math. Anal. Appl., 005, 30(: 5 3. [5] utwak E, Yang D, Zhang Gaoyong. Orlicz projection bodies[j]. Adv. Math., 00, 3(: 0 4. [6] utwak E, Yang D, Zhang Gaoyong. Orlicz centroid bodies[j]. J. Diff. Geom., 00, 84(: [7] Chen Fangwei, Zhou Jiazu, Yang Congli. On the reverse Orlicz Busemann-Petty centroid inequality[j]. Adv. Appl. Math., 0, 47(4:

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