Citation Journal of Inequalities and Applications, 2012, p. 2012: 90

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1 Title Polar Duals of Covex ad Star Bodies Author(s) Cheug, WS; Zhao, C; Che, LY Citatio Joural of Iequalities ad Applicatios, 2012, p. 2012: 90 Issued Date 2012 URL Rights This work is licesed uder a Creative Commos Attributio- NoCommercial-NoDerivatives 4.0 Iteratioal Licese.

2 Zhao et al. Joural of Iequalities ad Applicatios 2012, 2012:90 RESEARCH Polar duals of covex ad star bodies Chag-Jia Zhao 1*, Lia-Yig Che 1 ad Wig-Sum Cheug 2 Ope Access * Correspodece: chjzhao@yahoo. com.c 1 Departmet of Mathematics, Chia Jiliag Uiversity, Hagzhou , Chia Full list of author iformatio is available at the ed of the article Abstract I this article, some ew iequalities about polar duals of covex ad star bodies are established. The ew iequalities i special case yield some of the recet results. MR (2000) Subject Classificatio: 52A30. Keywords: polar dual, L p -mixed volume, dual L p -mixed volume, the Bourgai ad Milma s iequality 1 Notatios ad prelimiaries The settig for this article is -dimesioal Euclidea space R ( > 2). Let K deotes the set of covex bodies (compact, covex subsets with o-empty iteriors) i R. We reserve the letter u for uit vectors, ad the letter B for the uit ball cetered at the origi. The surface of B is S -l. The volume of the uit -ball is deoted by ω. We use V(K) forthe-dimesioal volume of covex body K. h(k, ) : S R, deotes the support fuctio of K K ; i.e., for u Î S -l h(k, u) = Max{u x : x K}, (1:1) where u xdeotes the usual ier product u ad x i R. Let δ deotes the Hausdorff metric o K,i.e.,forK, L K, δ(k, L) = h K h L, where deotes the sup-orm o the space of cotiuous fuctios C(S -l ). Associated with a compact subset K of R, which is star-shaped with respect to the origi, is its radial fuctio ρ(k, ) : S R, defied for u Î S -l,by ρ(k, u) =Max{λ 0:λu K}. (1:2) If r(k, ) is positive ad cotiuous, K will be called a star body. Let S deotes the set of star bodies i R.Let δ deotes the radial Hausdorff metric, as follows, if K, LÎ S, the δ(k, L) = ρ K ρ L (See [1,2]). 1.1 L p -mixed volume ad dual L p -mixed volume If K, L K, the L p -mixed volume V p (K, L) was defied by Lutwak (see [3]): V p (K, L) = 1 h(l, u) p ds p (K, u), (1:3) S where S p (K, ) deotes a positive Borel measure o S -1. The L p aalog of the classical Mikowski iequality (see [3]) states that: If K ad L are covex bodies, the 2012 Zhao et al; licesee Spriger. This is a Ope Access article distributed uder the terms of the Creative Commos Attributio Licese ( which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited.

3 Zhao et al. Joural of Iequalities ad Applicatios 2012, 2012:90 Page 2 of 5 V p (K, L) V(K) ( p)/ V(L) p/, (1:4) with equality if ad oly if K ad L are homothetic. If K, L Î S, p 1, the L p -dual mixed volume Ṽ p (K, L) was defied by Lutwak (see [4]): Ṽ p (K, L) = 1 +p ρ(l, u) p ds(u), (1:5) S ρ(k, u) where ds(u) sigifies the surface area elemet o S -1 at u. The followig dual L p -Mikowski iequality was obtaied i [2]: If K ad L are star bodies, the Ṽ p (K, L) V(K) +p V(L) p, (1:6) with equality if ad oly if K ad L are dilates. 1.2 Mixed bodies of covex bodies If K 1,..., K K, the otatio of mixed body [K 1,..., K -1 ] states that (see [5]): correspodig to the covex bodies K 1,..., K K i R, there exists a covex body, uique up to traslatio, which we deote by[k 1,..., K -1 ]. The followig is a list of the properties of mixed body: It is symmetric, liear with respect to Mikowski liear combiatios, positively homogeeous, ad for K i K, i =1,...,, L 1 K ad l i >0, (1) V 1 ([K 1,..., K -1 ], K )=V(K 1,..., K -1, K ); (2) [K 1 + L 1, K 2,..., K -1 ]=[K 1, K 2,..., K -1 ]+[L 1, K 2,..., K -1 ]; (3) [l 1 K 1,..., l -1 K -1 ]=l 1... l -1 [K 1,..., K -1 ]; [K,..., K] = K (4) }{{}. The properties of mixed body play a importat role i provig our mai results. 1.3 Polar of covex body For K K, the polar body of K, K* is defied: K = {x R : x y 1, y K}. It is easy to get that ρ(k, u) 1 = h(k, u). (1:7) Bourgai ad Milma s iequality is stated as follows (see [6]). If K is a covex symmetric body i R, the there exists a uiversal costat c>0 such that V(K)V(K ) c ω 2. (1:8) Differet proofs were give by Pisier [7].

4 Zhao et al. Joural of Iequalities ad Applicatios 2012, 2012:90 Page 3 of 5 2 Mai results I this article, we establish some ew iequalities o polar duals of covex ad star bodies. Theorem 2.1 If K, K 1,..., K -1 are covex bodies i R ad let L =[K 1,...,K -1 ], the the L p -mixed volumes V p (K, L), V p (K*, L), V p (B, L) satisfy V p (K, L)V p (K, L) V p (B, L) 2. (2:1) Proof From (1.1) ad (1.2), it is easy h(k, u) ρ(k, u), K K. (2:2) By defiitio of L p -mixed volume, we have V p (K, L) = 1 p ds p (L; u), (2:3) S h(k, u) ad V p (K, L) = 1 S h(k, u) p ds p (L, u). (2:4) Multiply both sides of (2.3) ad (2.4), i view of (1.7) ad (2.2) ad usig the Cauchy-Schwarz iequality (see [8]), we obtai 2 V p (K, L)V p (K, L) = h(k, u) p ds p (K 1,..., K ; u) 1 ρ(k, u) p ds p(k 1,..., K ; u) S S 2 p 1 h(k, u) 2 p ds p (K 1,..., K ; u) S ρ(k, u) 2 2 ds p (K 1,..., K ; u) S = 2 Vp 2 (B, L). Takig p = -1 i (2.1) ad i view of the property (1) of mixed body, we obtai the followig result: If K, K 1,..., K K, the V(K, K 1,..., K )V(K, K 1,..., K ) V(B, K 1,..., K ) 2. (2:5) This is just a iequality give by Ghadehari [9]. Let L = B, we have the followig iterestig result: Let K be a covex body ad K* its polar dual, the V p (K, B)V p (K, B) ω 2. (2:6) Takig p = -1 i (2.6), we have the followig result which was give i [9]: W (K)W (K ) ω 2,

5 Zhao et al. Joural of Iequalities ad Applicatios 2012, 2012:90 Page 4 of 5 with equality if ad oly if K is a -ball. Corollary 2.2 The L p -mixed volume of K ad K*, V p (K, K*) satisfies V p (K, K) ω 2( p) V(K) 2p. (2:7) Proof I view of the property (4) of the mixed body, we have V p (K,[K,..., K]) = V p (K, K) =V(K). Form (1.4) ad takig for K 1 = K 2 =... = K -1 = K i (2.1), we have V(K)V p (K, K) Vp 2 (B, K) V(B) 2( p) 2p V(K) = ω 2( p) V(K) 2p. Takig p = -1 i (2.7), we have the followig result: V(K, K,..., K) }{{} ω 2 V(K) 2. This is just a iequality give by Ghadehari [9]. The cases p = 1 ad =2give Steihardt s ad Firey s result (see [7]). A reverse iequality about Ṽ(K, K,..., K) }{{} was give by Ghadehari [9]. Ṽ(K, K,..., K) }{{} ω 2 V(K) 2. Theorem 2.3 Let K be a star body i R, K* be the polar dual of K, the there exist a uiversal costat c>0 such that V(K) +2p Ṽ p (K, K) (c ω 2 )+p, (2:8) where c is the costat of Bourgai ad Milma s iequality. Proof From (1.6) ad (1.8), we have Ṽ p (K, K) V(K ) =(V(K )V(K)) + p V(K) p + p +2p V(K) + p +2p (c ω 2 ) V(K). The followig theorem cocerig L p -dual mixed volumes will geeralize Sataló iequality. Theorem 2.4 Let K 1 ad K 2 be two star bodies, K1ad K2be the polar dual of K 1 ad K 2, the there exists a costat c, L p -dual mixed volumes Ṽ p (K 1, K 2 )ad Ṽ p (K 1, K 2 )Ṽ p (K 1, K 2 ) c ω 2.satisfy Ṽ p (K 1, K 2 )Ṽ p (K 1, K 2 ) c ω 2. (2:9)

6 Zhao et al. Joural of Iequalities ad Applicatios 2012, 2012:90 Page 5 of 5 Proof From (1.6), we have Ṽ p (K 1, K 2 ) (K 1 ) For K 1 ad K 2, we also have + p V(K 2 ) p. Ṽ p (K 1, K 2 ) V(K 1 ) + p V(K 2 ) p. (2:10) (2:11) Multiply both sides of (2.10) ad (2.11) ad usig Bourgai ad Milma s iequality, we obtai p p Ṽ p (K 1, K 2 )Ṽ p (K1, K 2 ) (V(K 1)V(K1 )) (V(K 2 )V(K2 )) + p p (c ω 2 ) (c ω 2 ) = c ω 2. Takig for K 1 =K 2 =Ki (2.9) ad i view of Ṽ p (K 1, K 2 )=Ṽ p (K, K) =V(K), (2.9) chages to the Bourgai ad Milma s iequality (1.8). Ackowledgemets C.-J. Zhao research was supported by Natioal Natural Scieces Foudatio of Chia ( ). W.-S. Cheug research was partially supported by a HKU URG grat. Author details 1 Departmet of Mathematics, Chia Jiliag Uiversity, Hagzhou , Chia 2 Departmet of Mathematics, The Uiversity of Hog Kog, Pokfulam Road, Hog Kog Authors cotributios C-JZ, L-YC ad W-SC joitly cotributed to the mai results Theorems 2.1, 2.3, ad 2.4. All authors read ad approved the fial mauscript. Competig iterests The authors declare that they have o competig iterests. Received: 17 December 2011 Accepted: 17 April 2012 Published: 17 April 2012 Refereces 1. Scheider, R: Covex Boides: The Bru-Mikowski Theory. Cambridge Uiversity Press Cambridge (1993) 2. Garder, RJ: Geometric Tomography. Cambridge Uiversity Press New York (1996) 3. Lutwak, E: The Bru-Mikowski-Firey theory-i: mixed volumes ad the Mikowski problem. J Diff Geom. 38, (1993) 4. Lutwak, E, Yag, D, Zhag, GY: L p affie isoperimetric iequalities. J Diff Geom. 56, (2000) 5. Lutwak, E: Volume of mixed bodies. Tras Am Math Soc. 294, (1986) 6. Bourgai, J, Milma, V: New volume ratio properties for covex symmetric bodies i [ieq]. Ivet Math. 88, (1987) 7. Pisier, G: The volume of covex bodies ad Baach space geomery. Cambridge Uiversity Press Cambridge (1989) 8. Hardy, GH, Littlewood, JE, Pólya, G: Iequalities. Cambridge Uiversity Press Cambridge (1934) 9. Ghadehari, M: Polar duals of covex bodies. Proc Am Math Soc. 113(3): (1991) doi: / x Cite this article as: Zhao et al.: Polar duals of covex ad star bodies. Joural of Iequalities ad Applicatios :90.