Addition some inequality of volumes mixed in geometry

The technique we adopt is the solution for the function Minkowski problem on W 1,p(Rn) established by Lutwak, Yang and Zhang [8]. This technique plays a fundamental role in the newly emerged affine Sobolev inequalities (see, e.g., [6, 8, 9]). The importance of this technique is that it convert analytic inequalities to geometry inequalities. Note that the geometry behind the sharp Lp Sobolev inequality is the isoperimetric inequality, while the affine Lp Sobolev inequality is equivalent to the Lp Petty projection inequality established in [10, 11].

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2015-0026 Mathematical and Physical Sci., 2015, Vol. 60, No. 7, pp. 15-20 This paper is available online at ADDITION SOME INEQUALITY OF VOLUMES MIXED IN GEOMETRY Lai Duc Nam Yen Bai Teacher’s Training College, Yen Bai Abstract. The issue norms for all volume preserving affine transformations for optimal Sobolev is always been interested in research. In this paper, we present some results of inequalities of volume mixed in geometry. Keywords: Lp Sobolev inequality, geometry inequalities. 1. Introduction The classical sharp Lp Sobolev inequality states that if f ∈ W 1,p(Rn), with real p satisfying 1 ≤ p < n, then ‖∇f‖p ≥ αn,p‖f‖ np n−p (1.1) where ‖ · ‖q denotes the usual Lq norm for functions on Rn. The optimal constants αn,p in this inequality are due to Federer and Fleming [1] for p = 1 and to Aubin [2] for 1 < p < n. For strengthened versions of (1.1), see, e.g. [3, 4, 5] and the references therein. Recently, Zhang [6] (for p = 1) and Lutwak, Yang, and Zhang [7] (for 1 < p < n) formulated and proved a sharp affine Lp Sobolev inequality. This remarkable inequality is invariant under all affine transformations of Rn, while the classical Lp Sobolev inequality (1.1) is invariant only under rigid motions. In the affine Lp Sobolev inequality the Lp norm of the Euclidean length of the gradient is replaced by an affine invariant of functions, the Lp affine energy, defined, for f ∈W 1,p(Rn), by Ep(f) = cn,p (∫ Sn−1 ‖Duf‖−np du )−1/n (1.2) where cn,p = (nκn)1/n( nκnκp−1 2κn+p−2 )1/p with κn = πn/2/Γ(1 + n 2 ) and Duf is the directional derivative of f in the direction u. The sharp affine Lp Sobolev inequality of Zhang [6] and Lutwak, Yang, and Zhang [7] states that if f ∈W 1,p(Rn), 1 ≤ p < n, then Ep(f) ≥ αn,p‖f‖ np n−p Received October 10, 2015. Accepted November 30, 2015. Contact Lai Duc Nam, e-mail address: nam.laiduc@gmail.com 15 Lai Duc Nam It is shown in [7;33] that ‖∇f‖p ≥ Ep(f). (1.3) Hence the sharp affine Lp Sobolev inequality (1.2) is stronger than the classical affine Lp Sobolev inequality (1.1). We emphasize the remarkable and important fact that Ep(f) is invariant under volume preserving affine transformations on Rn. In contrast, ‖∇f‖p is invariant only under rigid motions. We also show a reverse inequality of (1.3), there exist a T˜p ∈ SL(n) such that cp‖∇(f ◦ T˜p)‖p ≤ Ep(f), where cp is a constant only depending on p. The technique we adopt is the solution for the function Minkowski problem on W 1,p(Rn) established by Lutwak, Yang and Zhang [8]. This technique plays a fundamental role in the newly emerged affine Sobolev inequalities (see, e.g., [6, 8, 9]). The importance of this technique is that it convert analytic inequalities to geometry inequalities. Note that the geometry behind the sharp Lp Sobolev inequality is the isoperimetric inequality, while the affine Lp Sobolev inequality is equivalent to the Lp Petty projection inequality established in [10, 11]. 2. Notations and preliminaries We work in Rn, which is equipped with a Euclidean structure x · y for x, y ∈ Rn. We write Bn2 for the Euclidean unit ball and S n−1 for the unit sphere. A convex body is a compact convex set in Rn which is throughout assumed to contain the origin in its interior. We denote by Kno the space of convex bodies equipped with the Hausdorff metric. Each convex bodyK is uniquely determined by its support function hK = h(K, ·) : Rn → R defined by hK(x) = h(K,x) := max{x · y : y ∈ K}. Let ‖ · ‖K : Rn → [0,∞) denote the Minkowski functional of K ∈ Kno ; i.e., ‖x‖K = min{λ ≥ 0 : x ∈ λK}. The polar set K∗ ofK ∈ Kno is the convex body defined by K∗ = {x ∈ Rn : x · y ≤ 1 for all y ∈ K}. If K ∈ Kno , then it follows from the definitions of support functions and Minkowski functionals, and the definition of polar body, that h(K∗, ·) = ‖ · ‖K . For p ≥ 1,K,L ∈ Kno , the Lp Minkowski combination K +p L is the convex body defined by h(K +p L, ·)p = h(K, ·)p + h(L, ·)p. Introduced by Firey in the 1960’s, this notion is the basis of what has become known as the Lp Brunn-Minkowski theory (or the Brunn-Minkowski-Firey theory). 16 Addition some inequality of volumes mixed in geometry The Lp mixed volume Vp(K,L) of K,L ∈ Kno is defined in [12] by Vp(K,L) = p n lim ε→0+ V (K +p ε 1 pL)− V (K) ε . In particular, Vp(K,K) = V (K) for every convex body K . It was shown in [12] that corresponding to each convex body K ∈ Kno , there is a positive Borel measure on Sn−1, the Lp surface area measure Sp(K, ·) ofK , such that for every L ∈ Kno , Vp(K,L) = 1 n ∫ Sn−1 hpL(u)dSp(K,u). The measure S1(K, ·) is just the surface area measure ofK . Recall that for a Borel set ω ⊂ Sn−1, the surface area measure S(K,ω) 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. Moreover, the Lp surface area measure is absolutely continuous with respect to S(K, ·): dSp(K,u) = hK(u) 1−pdS(K,u) u ∈ Sn−1. Note that Sp(tK, ·) = tn−pSp(K, ·) (2.1) for all t > 0 and convex bodies K . 3. Proof of main results Following the definition of Lutwak, Yang and Zhang [8], we define the normalized Lp projection body Π˜pK ofK by h(Π˜pK,u) p = 1 V (K) ∫ Sn−1 |u · v|pdSp(K, v). (3.1) Theorem 3.1. LetK ∈ Kno and p ≥ 1, then V (Π˜∗pK) ≥ κn (nκp−1κnV (K) 2κn+p−2Sp(K) )n p . Proof. Using (3.1), the polar coordinate formula for volume, Ho¨lder inequality and Fubini 17 Lai Duc Nam theorem, we obtain V (Π˜∗pK) = 1 n ∫ Sn−1 h(Π˜pK,u) −ndu ≥ κ1+ n p n ( 1 n ∫ Sn−1 h(Π˜pK,u) pdu )−n p = κ 1+n p n ( 1 nV (K) ∫ Sn−1 ∫ Sn−1 |u · v|pdSp(K, v)du )−n p = κ 1+n p n ( 1 nV (K) ∫ Sn−1 ∫ Sn−1 |u · v|pdudSp(K, v) )−n p = κ 1+n p n ( 2κn+p−2 nκp−1V (K) ∫ Sn−1 dSp(K, v) )−n p = κn (nκp−1κnV (K) 2κn+p−2Sp(K) )n p Theorem 3.2. LetK ∈ Kno and p ≥ 1. If Sp(K, ·) is isotropic, then( Sp(K) nV (K) ) 1 p ≤ h(Π˜pK,u) ≤ 1√ n ( Sp(K) nV (K) ) 1 p , 1 ≤ p ≤ 2, and ( Sp(K) nV (K) ) 1 p ≥ h(Π˜pK,u) ≥ 1√ n ( Sp(K) nV (K) ) 1 p , p ≥ 2, for every u ∈ Sn−1. Proof. For the case that p = 2, by the definition of the normalized Lp projection body (3.1), we have h(Π˜2K,u) = √ S2(K)/nV (K). For 1 ≤ p ≤ 2, by (3.1) and the Ho¨lder inequality, we obtain h(Π˜pK,u) = ( 1 V (K) ∫ Sn−1 |u · v|pdSp(K, v) ) 1 p ≤ [( 1 V (K) ∫ Sn−1 |u · v|2dSp(K, v) ) p 2 Sp(K) 1− p 2 ] 1 p = 1√ n ( Sp(K) nV (K) ) 1 p for every u ∈ Sn−1. On the other hand, we have h(Π˜pK,u) ≥ ( 1 V (K) ∫ Sn−1 |u · v|2dSp(K, v) ) 1 p = ( Sp(K) nV (K) ) 1 p for every u ∈ Sn−1. Similarly, we can get the result for p ≥ 2. 18 Addition some inequality of volumes mixed in geometry In 2006, Lutwak, Yang and Zhang [8] proposed and proved the following functional Minkowski problem onW 1,p(Rn). Theorem 3.3. Given 1 ≤ p < ∞ and a function f ∈ W 1,p(Rn), there exists a unique origin-symmetric convex body 〈f〉p that∫ Rn Φ(−∇f(x))pdx = 1 V (〈f〉p) ∫ Sn−1 Φ(v)pdSp(〈f〉p, v), (3.2) for every even continuous function Φ : Rn → [0,∞) that is homogeneous of degree 1. Like the known result that the affine Lp Sobolev inequality implies the Lp Petty projection inequality, we will show that our main theorems implies corresponding geometry theorems listed in [13]. Similar to the proof of [14, Lemma 3], we show the following theorem. Theorem 3.4. If K ∈ Kno and f = g(‖x‖K) with g ∈ C1(0,∞), then for p ≥ 1, 〈f〉p is a dilate of K , that is 〈f〉p = c(f)− 1 p K˜ where c(f) = ∫∞ 0 t n−1|g′(t)|pdt and K˜ = K/|K|. Proof. Since hK∗ is Lipschitz (and therefore differentiable almost everywhere) and hK∗(x) = 1 on ∂K , then for almost every x ∈ ∂K , νK(x) = ∇hK∗(x) |∇hK∗(x)| , where νK(x) is the outer unit normal vector ofK at the point x. Note that hK(∇hK∗(x)) = 1, for almost every x ∈ Rn. Hence we have hK(νK(x)) = 1 |∇hKo(x)| . (3.3) Then by (3.2), the co-area formula (see, e.g., [15;258]) applied to hK∗(·), the fact that ∇hK˜∗ is homogeneous of degree 0 and (3.3), we get 1 V (〈f〉p) ∫ Sn−1 Φp(u)dSp(〈f〉p, u) = ∫ Rn Φp(∇f(x))dx = ∫ Rn Φp(g′(hK∗(x))∇hK∗(x)dx = ∫ ∞ 0 ∫ ∂K tn−1|g′(t)|pΦp ( ∇hK∗(x) |∇hK∗(x)| ) |∇hKo(x)|p−1dHn−1(x)dt = ∫ ∞ 0 tn−1|g′(t)|pdt ∫ ∂K Φp(νK(x))hK(νK(x)) 1−pdHn−1(x) = ∫ ∞ 0 tn−1|g′(t)|pdt ∫ Sn−1 Φp(u)dSp(K,u) 19 Lai Duc Nam for every even continuous function Φ that is homogeneous of degree 1. Thus, the uniqueness of the solution of the even Lp Minkowski problem [12] and (2.1) imply that 〈f〉p V (〈f〉p) 1 n−p = c(f) 1 n−pK, and hence 〈f〉p = c(f)− 1 p K˜ where c(f) = ∫∞ 0 t n−1|g′(t)|pdt and K˜ = K/V (K)1/p. REFERENCES [1] H. Federer, W. Fleming, 1960. Normal and integral currents. Ann. Math. 72, 458-520. [2] T. Aubin, 1976. Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11, 573-598. [3] D. Cordero-Erausquin, D. Nazaret, C. Villani, 2004. A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182, 307-332. [4] M. Del Pino, J. Dolbeault, 2002. Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl. 81, 847-875. [5] J. Xiao, 2007. The sharp Sobolev and isoperimetric inequalities split twice. Adv. Math. 211, 417-435. [6] G. Zhang, 1999. The affine Sobolev inequality. J. Differential Geom. 53, 183-202. [7] E. Lutwak, D. Yang, G. Zhang, 2002. Sharp affine Lp Sobolev inequalities. J. Differential Geom. 62, 17-38. [8] E. Lutwak, D. Yang, G. Zhang, 2006.Optimal Sobolev norms and the Lp Minkowski problem. Int. Math. Res. Not. Art. ID 62987, 21 pp. [9] D. Alonso-Gutierrez, J. Bastero, J. Bernués, 2012. Factoring Sobolev inequalities through classes of functions. Proc. Amer. Math. Soc. 140, 3557-3566. [10] S. Campi, P. Gronchi, 2002. The Lp-Busemann-Petty centroid inequality. Adv. Math. 167, 128-141. [11] E. Lutwak, D. Yang, G. Zhang, 2000. Lp affine isoperimetric inequalities. J. Differential Geom. 56, 111-132. [12] E. Lutwak, 1993. The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem. J. Differential Geom. 38, 131-150. [13] W. Yu, 2011. Isotropic p-surface area measure. Acta Math. Sci. Ser. A Chin. Ed. 31, 644-651. [14] T. Wang, 2013. The affine Pólya-Szego¨ principle: Equality cases and stability. J. Funct. Anal. 265, 1728-1748. [15] H. Federer, 1969. Geometric Measure Theory. Springer, Berlin. 20