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 THE ORIGIN SYMMETRIC FOR POLYTOPES Bui Thi Nghia1, Tran Thi Hien2, Dinh Thi Van Khanh2 and Lai Duc Nam2 1Hoang Quoc Viet Upper Secondary School, Yenbai 2Yen 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 - Su¨ss 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 polytopesK and L inRn, a new convex polytope K♯L, called the Blaschke sum ofK 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 theLp-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-Su¨ss inequality. Furthermore, an application of Lp Kneser-Su¨ss 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 Pn denote the subset of convex polytopes. The members of Kn are called convex bodies. We 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 Kno for the set of convex bodies which contain the origin as an interior point, and put Pno = Pn ∩ Kno . 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〈x, y〉, where 〈x, y〉 is the standard inner product of x and y in Rn. We shall use V (K) to denote n-dimensional volume of a convex bodyK 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 : 〈x, u〉 = h(K,u). The (n−1)-dimensional support sets of a polytope P ∈ Pn are called the facets of P . If P ∈ Pn has facets F (P, ui) with areas ai, i = 1, · · · ,m, then S(P, ·) is the discrete measure S(P, ·) = m∑ i=1 aiδi 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 ⊂ Sn−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 ofK belonging to ω. For p ≥ 1, it was shown in [4] that corresponding to each convex body K ∈ Kno , there is a positive Borel measure on Sn−1, the Lp surface area measure Sp(K, ·) of K , such that for everyL ∈ Kno , Vp(K,L) = 1 n ∫ Sn−1 h(L, u)pdSp(K,u). (2.1) Moreover, Vp(K,K) = V (K). The measure S1(K, ·) is just the surface area measure ofK . Moreover, the Lp surface area measure is absolutely continuous with respect to S(K, ·): Sp(K, ·) = h(K, ·)1−pS(K, ·). The Lp Minkowski inequality [4] states: IfK ∈ Kno , then Vp(K,L) n ≥ V (K)n−pV (L)p (2.2) with equality if and only ifK 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 ∈ Sn−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 ∈ Pno such that m∑ j=1 αujδuj = h(P, ·)1−pS(P, ·). 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 ∈ Pno , n 6= p ≥ 1, the Lp-Blaschke addition K♯pL ∈ Pno ofK and L is defined by Sp(K♯pL, ·) = Sp(K, ·) + Sp(L, ·). (3.1) Note that the Lp-Blaschke addition for K,L ∈ Kne is previously defined by Lutwak [4], who also obtained the following Lp Kneser-Su¨ss inequality for K,L ∈ Kne . By the same arguments, we obtain the Lp Kneser-Su¨ss inequality forK,L ∈ Pno . Theorem 3.1. IfK,L ∈ Pno , and 1 < p 6= n, then V (K♯pL) (n−p)/n ≥ V (K)(n−p)/n + V (L)(n−p)/n, with equality if and only ifK and L are dilates. Proof. From (2.1) and (3.1), we have Vp(K♯pL,Q) = Vp(K,Q) + Vp(L,Q). Together with (2.2) yields Vp(K♯pL,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♯pL 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 0 such that (3.2) fails. Theorem 3.2. For every a ≥ 1 and K,L ∈ Pno (1 < p 6= n, n ≥ 2), the following inequalities hold: V (K♯pL) (an−ap)/n ≥ V (K)(an−ap)/n + V (L)(an−ap)/n. (3.3) Moreover, for 0 < a < 1 there exist K,L ∈ Pno such that the inequality (3.3) fail. Proof. For a ≥ 1, by Theorem 3.1 and the inequality (3.2), we have[V (K♯pL) V (K) ]a(n−p)/n ≥ {1 + [ V (L) V (K) ](n−p)/n}a ≥ 1 + [ V (L) V (K) ]a(n−p)/n , 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♯pL, ei) 1−pS(K♯pL, ei) = h(K, ei)1−pS(K, ei) + h(L, ei)1−pS(L, ei) = r1−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♯pL = tZ for some t > 0. Moreover, h(K♯pL, ei) 1−pS(K♯pL, ei) = h(tZ, ei)1−pS(tZ, ei) = tn−p2n−1 = r1−p(2r)n−1 +R1−p(2R)n−1. Consequently, we have K♯pL = tZ = (rn−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 (rn−p +Rn−p)a ≥ ra(n−p) +Ra(n−p),( 1 + ( r R )(n−p))a ≥ 1 + ( r R )a(n−p) . However, by Lemma 3.2 for 0 < a < 1, the latter inequality certainly fails for some x = (r/R)n−p. Theorem 3.3. LetK,L ∈ Pno (n ≥ 2) and let the Lp surface area (1 < p 6= n)measure ofK 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 Sn−1, we conclude that µ is not contained in a closed hemisphere of Sn−1. By Lemma 3.1, there exists a convex bodyM 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(t 1/(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 ∈ Kno 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]. 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