The origin symmetric for polytopes

Chia sẻ: Y Y | Ngày: | Loại File: PDF | Số trang:5

Thêm vào BST

Báo xấu

8
lượt xem 1
download

Download Vui lòng tải xuống để xem tài liệu đầy đủ

The Lp-Blascke addition for a pair of origin symmetric convex bodies is extended to a pair of convex polytopes containing the origin in their interior and Lp Kneser - Suss inequality for polytopes is established.

Chủ đề:

Bình luận(0) Đăng nhập để gửi bình luận!

Lưu

Nội dung Text: The origin symmetric for polytopes

JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2015-0028 Mathematical and Physical Sci., 2015, Vol. 60, No. 7, pp. 30-34 This paper is available online at http://stdb.hnue.edu.vn THE ORIGIN SYMMETRIC FOR POLYTOPES Bui Thi Nghia1 , Tran Thi Hien2 , Dinh Thi Van Khanh2 and Lai Duc Nam2 1 Hoang Quoc Viet Upper Secondary School, Yenbai 2 Yen Bai Teacher’s Training College, Yen Bai Abstract. The Lp -Blascke addition for a pair of origin symmetric convex bodies is extended to a pair of convex polytopes containing the origin in their interior and Lp Kneser - S¨ uss inequality for polytopes is established. Keywords: Convex body, Blascke addition, Lp -Blascke addition, polytopes. 1. Introduction The operation between convex bodies now called Blaschke addition goes back to Minkowski [1;117], at least when the bodies are polytopes. Given convex polytopes K and L in Rn , a new convex polytope K♯L, called the Blaschke sum of K and L, has a facet with a normal outer unit in a given direction if and only if either K or L (or both) do, in which case the area (i.e., (n − 1)-dimensional volume) of the facet is the sum of the areas of the corresponding facets of K and L. Blaschke [2;112] found a definition suitable for smooth convex bodies in R3 . The modern definition, appropriate for any pair of convex bodies, had to wait for the development of surface area measures and is due to Fenchel and Jessen [3]. They defined the surface area measure of K♯L to be the sum of the surface area measures of K and L, and this determines the Blaschke sum, up to translation (See [1]). The existence of K♯L is guaranteed by Minkowski’s existence theorem, a classical result that can be found, along with definitions and terminology, in Section 2. The Lp -Blaschke addition for any pair of origin symmetric convex bodies was defined by Lutwak [4], by using the solution of the even Lp Minkowski problem. In this paper, we extend the Lp -Blaschke addition to any pair of convex polytopes containing the origin in their interior from a pair of origin symmetric convex bodies. From this definition, we extend Lutwak’s Lp Kneser-S¨uss inequality. Furthermore, an application of Lp Kneser-S¨uss inequality is presented. 2. Notations and preliminaries For general reference, the reader may wish to consult the books of Gardner [5], Schneider [6]. Let Kn denote the space of compact convex subsets of Rn with nonempty interiors, and let P denote the subset of convex polytopes. The members of Kn are called convex bodies. We n Received July 25, 2015. Accepted November 24, 2015. Contact Lai Duc Nam, e-mail address: nam.laiduc@gmail.com 30
The origin symmetric for polytopes write Kon for the set of convex bodies which contain the origin as an interior point, and put Pon = P n ∩ Kon . For a convex body K let hK = h(K, ·) : Rn → R denote the support function of K; i.e., for x ∈ Rn , let hK (x) = maxy∈K hx, yi, where hx, yi is the standard inner product of x and y in Rn . We shall use V (K) to denote n-dimensional volume of a convex body K in Rn . For K ∈ Kn , let F (K, u) denote the support set of K with exterior unit normal vector u, i.e. F (K, u) = x ∈ K : hx, ui = h(K, u). The (n−1)-dimensional support sets of a polytope P ∈ P n are called the facets of P . If P ∈ P n has facets F (P, ui ) with areas ai , i = 1, · · · , m, then S(P, ·) is the discrete measure m X S(P, ·) = ai δi i=1 with (finite) support {u1 , · · · , um } and S(P, {ui }) = ai , i = 1, · · · , m; here δi denotes the probability measure with unit point mass at ui . For a Borel set ω ⊂ S n−1 , the surface area measure SK (ω) = S(K, ω) of the convex body K is the (n − 1)-dimensional Hausdorff measure of the set of all boundary points of K for which there exists a normal vector of K belonging to ω. For p ≥ 1, it was shown in [4] that corresponding to each convex body K ∈ Kon , there is a positive Borel measure on S n−1 , the Lp surface area measure Sp (K, ·) of K, such that for everyL ∈ Kon , Z 1 Vp (K, L) = h(L, u)p dSp (K, u). (2.1) n S n−1 Moreover, Vp (K, K) = V (K). The measure S1 (K, ·) is just the surface area measure of K. Moreover, the Lp surface area measure is absolutely continuous with respect to S(K, ·): Sp (K, ·) = h(K, ·)1−p S(K, ·). The Lp Minkowski inequality [4] states: If K ∈ Kon , then Vp (K, L)n ≥ V (K)n−p V (L)p (2.2) with equality if and only if K and L are dilates. 3. Proof of main results In [7], Hug et al. established the solution to the discrete-data case of the Lp Minkowski problem. Lemma 3.1. Let vectors u1 , · · · , um ∈ S n−1 that are not contained in a closed hemisphere and real numbers α1 , · · · , αm > 0 be given. Then, for any p > 1 with p 6= n, there exists a unique polytope P ∈ Pon such that X m αuj δuj = h(P, ·)1−p S(P, ·). j=1 31
Bui Thi Nghia, Tran Thi Hien, Dinh Thi Van Khanh and Lai Duc Nam From this theorem, we can define the Lp -Blaschke addition: for K, L ∈ Pon , n 6= p ≥ 1, the Lp -Blaschke addition K♯p L ∈ Pon of K and L is defined by Sp (K♯p L, ·) = Sp (K, ·) + Sp (L, ·). (3.1) Note that the Lp -Blaschke addition for K, L ∈ Ken is previously defined by Lutwak [4], who also obtained the following Lp Kneser-S¨uss inequality for K, L ∈ Ken . By the same arguments, we obtain the Lp Kneser-S¨uss inequality for K, L ∈ Pon . Theorem 3.1. If K, L ∈ Pon , and 1 < p 6= n, then V (K♯p L)(n−p)/n ≥ V (K)(n−p)/n + V (L)(n−p)/n , with equality if and only if K and L are dilates. Proof. From (2.1) and (3.1), we have Vp (K♯p L, Q) = Vp (K, Q) + Vp (L, Q). Together with (2.2) yields Vp (K♯p L, Q) ≥ V (Q)p/n [V (K)(n−p)/n + V (L)(n−p)/n ] with equality (for p > 1) if and only if K, L and Q are dilates. The result follows by taking K♯p L for Q. Lemma 3.2. For all a ≥ 1 and x > 0 the following inequality holds: (1 + x)a ≥ 1 + xa . (3.2) Moreover, for all 0 < a < 1 there exists x > 0 such that (3.2) fails. Theorem 3.2. For every a ≥ 1 and K, L ∈ Pon (1 < p 6= n, n ≥ 2), the following inequalities hold: V (K♯p L)(an−ap)/n ≥ V (K)(an−ap)/n + V (L)(an−ap)/n . (3.3) Moreover, for 0 < a < 1 there exist K, L ∈ Pon such that the inequality (3.3) fail. Proof. For a ≥ 1, by Theorem 3.1 and the inequality (3.2), we have h V (K♯ L) ia(n−p)/n n h V (L) i(n−p)/n oa h V (L) ia(n−p)/n p ≥ 1+ ≥1+ , V (K) V (K) V (K) that proves (3.3) for a ≥ 1. Now, let 0 < a < 1. Let Z = [−1, 1]n−1 , K = rZ, L = RZ. Then, for i = 1, · · · , n, h(K, ei ) = r h(L, ei ) = R, S(K, ei ) = (2r)n−1 S(K, ei ) = (2R)n−1 . 32
The origin symmetric for polytopes Thus, h(K♯p L, ei )1−p S(K♯p L, ei ) = h(K, ei )1−p S(K, ei ) + h(L, ei )1−p S(L, ei ) = r 1−p (2r)n−1 + R1−p (2R)n−1 . Note that for v 6= ei i = 1, · · · , n, S(K, v) = 0 and S(L, v) = 0. Thus, K♯p L = tZ for some t > 0. Moreover, h(K♯p L, ei )1−p S(K♯p L, ei ) = h(tZ, ei )1−p S(tZ, ei ) = tn−p 2n−1 = r 1−p (2r)n−1 + R1−p (2R)n−1 . Consequently, we have K♯p L = tZ = (r n−p + Rn−p )1/(n−p) Z. From (3.3), we have V (tZ)a(n−p)/n ≥ V (rZ)a(n−p)/n + V (RZ)a(n−p)/n or (r n−p + Rn−p )a ≥ r a(n−p) + Ra(n−p) , r (n−p) a r a(n−p) 1+ ≥1+ . R R However, by Lemma 3.2 for 0 < a < 1, the latter inequality certainly fails for some x = (r/R)n−p . Theorem 3.3. Let K, L ∈ Pon (n ≥ 2) and let the Lp surface area (1 < p 6= n) measure of K does not exceed the surface area measure of L, that is Sp (K, ·) ≤ Sp (L, ·). Then, V (K) ≤ V (L), for 1 < p < n V (K) ≥ V (L), for p > n. Proof. Take t ∈ (0, 1). Consider an additive set function µ(·) of the unit sphere µ(·) such that µ(·) = Sp (L, ·) − tSp (K, ·). Since S(K, ·) is not contained in a closed hemisphere of S n−1 , we conclude that µ is not contained in a closed hemisphere of S n−1 . By Lemma 3.1, there exists a convex body M in Rn whose surface area function coincides with µ, that is µ(·) = Sp (M, ·). But then Sp (L, ·) = Sp (M, ·) + tSp (K, ·), and so L = M ♯p (t1/(n−p) K). By Theorem 3.1, we then have V (L)(n−p)/n = V (M ♯p (t1/(n−p) K)(n−p)/n ≥ V (M )(n−p)/n + V (t1/(n−p) K)(n−p)/n = V (M )1/(n−p) + t(V (K))(n−p)/n ≥ tV (K)(n−p)/n Tending t to 1, we get V (L)(n−p)/n ≥ V (K)(n−p)/n , which completes the proof. 33
Bui Thi Nghia, Tran Thi Hien, Dinh Thi Van Khanh and Lai Duc Nam Remark: The Lp Blaschke addition (3.1) in this paper also applies to K ∈ Kon for p > n. In this case, the existence of the Lp Blaschke addition is guaranteed by the solution of the general Lp Minkowski problem [7]. However, for 1 < p < n, the origin may lie on the boundary of K by the solution of the general Lp Minkowski problem, which will lead to h(K, v) = 0 for some v ∈ S n−1 . REFERENCES [1] H. Minkowski, 1897. Allgemeine Lehrs¨atze u¨ ber die convexen Polyeder. Nachr. Ges. Wiss. G¨ottingen, pp. 198-219. Gesammelte Abhandlungen. Vol. II. (Teubner, Leipzig, 1911), pp. 103-121. [2] W. Blaschke, 1956. Kreis und Kugel, second edition. W. de Gruyter, Berlin. [3] W. Fenchel and B. Jessen, 1938. Mengenfunktionen und konvexe K¨orper. Danske Vid. Selsk. Math.-Fys. Medd. Vol. 16, No. 3, p. 31. [4] E. Lutwak, 1993. The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem. J. Differential Geom, Vol. 38, pp. 131-150. [5] R. J. Gardner, 2006. Geometric Tomography, second edition. Cambridge University Press, New York. [6] R. Schneider, 1993. Convex Bodies: the Brunn-Minkowski Theory. Cambridge: Cambridge University Press. [7] D. Hug, E. Lutwak, D. Yang and G. Zhang, 2005. On the Lp Minkowski problem for polytopes. Discrete and Computational Geometry, Vol. 33, pp. 699-715. 34