ON ANALOGS OF THE DUAL BRUNN-MINKOWSKI INEQUALITY FOR WIDTH-INTEGRALS OF CONVEX BODIES

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1 STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LII, Number 2, Jue 2007 ON ANALOGS OF THE DUAL BRUNN-MINKOWSKI INEQUALITY FOR WIDTH-INTEGRALS OF CONVEX BODIES ZHAO CHANGJIAN, WING-SUM CHEUNG, AND MIHÁLY BENCZE Abstract. I this paper we prove two ew iequalities about widthitegrals of cetroid ad projectio bodies. Two aalogs of the dual Bru- Mikowski iequality for width-itegral of covex bodies are established. 0. Defiitios ad prelimiary results The settig for this paper is -dimesioal Euclidea space R ( > 2). Let K deote the set of covex bodies (compact, covex subsets with o-empty iteriors) i R. Let ϕ deote the set of star bodies i R. The subset of ϕ cosistig of the cetred star bodies will be deoted by ϕ c. We reserve the letter u for uit vectors, ad the letter B is reserved for the uit ball cetered at the origi. The surface of B is S. For u S, let E u deote the hyperplae, through the origi, that is orthogoal to u. We will use K u to deote the image of K uder a orthogoal projectio oto the hyperplae E u. We use V (K) for the -dimesioal volume of covex body K. Let h(k, ) : S R, deote the support fuctio of K K ; i.e. h(k, u) Max{u x : x K}, u S, () where u x deotes the usual ier product u ad x i R. Let δ deote the Hausdorff metric o K ; i.e., for K, L K, δ(k, L) h K h L, Received by the editors: Mathematics Subject Classificatio. 52A40. Key words ad phrases. width-itegral, projectio body, cetroid body, Blaschke liear combiatio, harmoic Blaschke liear combiatio. 3

2 ZHAO CHANGJIAN, WING-SUM CHEUNG, AND MIHÁLY BENCZE where deotes the sup-orm o the space of cotiuous fuctios, C(S ). For a covex body K ad a oegative scalar λ, λk, is used to deote {λx : x K}. For K i K, λ i 0, (i, 2,..., r),the Mikowski liear combiatio λ K + + λ r K r K is defied by λ K + + λ r K r {λ x + + λ r x r K : x i K i }. (2) It is trivial to verify that h(λ K + + λ r K r, ) λ h(k, ) + + λ r h(k r, ). (3). Mixed volumes If K i K (i, 2,..., r) ad λ i (i, 2,..., r)are oegative real umbers, the of fudametal impotece is the fact that the volume of λ K + + λ r K r is a homogeeous polyomial i λ i give by [4,p.275] V (λ K + + λ r K r ) i,...,i λ i... λ i V i...i, (4) where the sum is take over all -tuples (i,..., i ) of positive itegers ot exceedig r. The coefficiet V i...i depeds oly o the bodies K i,..., K i, ad is uiquely determied by (8), it is called the mixed volume of K i,..., K i, ad is writte as V (K i,..., K i ). Let K... K K ad K +... K L, the the mixed volume V (K... K ) is usually writte V i (K, L). If L B, the V i (K, B) is the ith projectio measure(quermassitegral) of K ad is writte as W i (K). If K i (i, 2,..., ) K, the the mixed volume of the covex figures K u i (i, 2,..., ) i the ( )-dimesioal space E u will be deoted by v(k u,..., K u ). It is well kow, ad easily show [5,p.45], that for K i K (i, 2,..., ), ad u S v(k u,..., K u ) V (K,..., K, [u]) (5) where [u] deotes the lie segmet joiig u/2 ad u/2..2 Width-itegrals of covex bodies For u S, b K 2 (h(k, u) + h(k, u)) is called as half the width of K i the directio u. Two covex bodies K ad L are said to have similar width if there exists 4

3 ON ANALOGS OF THE DUAL BRUNN-MINKOWSKI INEQUALITY a costat λ > 0 such that b K λb L for all u S. The width-itegral of idex i is defied by: For K K, i R B i (K) b K S where ds is the ( )-dimesioal volume elemet o S. The width-itegral of idex i is a map B i : K R. ds(u), (6) It is positive, cotiuous, homogeeous of degree i ad ivariat uder motio. I additio, for i it is also bouded ad mootoe uder set iclusio. The followig result easy is proved, for K j K (j,..., m) b K+ +K m b K + + b Km, (7).3 The Blaschke liear combiatio ad the harmoic Blaschke liear combiatio A covex body K is said to have a positive cotiuous curvature fuctio [8], f(k, ) : S [0, ), if for each L ϕ, the mixed volume V (K, L) has the itegral represetatio V (K, L) f(k, u)h(l, u)ds(u). S The subset of K cosistig of bodies which have a positive cotiuous curvature fuctio will be deoted by κ. Let κ c deote the set of cetrally symmetric member of κ. The followig result is true [9], for K κ S uf(k, u)ds(u) 0. Suppose K, L κ ad λ, µ 0(ot both zero). From above it follows that the fuctio λf(k, ) + µf(l, ) satisfies the hypothesis of Mikowski s existece theorem (see [5]). λ K +µ L that is The solutio of the Mikowski problem for this fuctio is deoted by f(λ K +µ L, ) λf(k, ) + µf(l, ), (8) 5

4 ZHAO CHANGJIAN, WING-SUM CHEUNG, AND MIHÁLY BENCZE where the liear combiatio λ K +µ L is called a Blaschke liea combiatio. The relatioship betwee Blaschke ad Mikowski scalar multiplicatio is give by λ K λ /( ) K. (9) A ew additio, harmoic Blaschke additio, be defied by Lutwak [8]. Suppose K, L ϕ, ad λ, µ 0(ot both zero). To defie the harmoic Blaschke liear combiatio, λk ˆ+µL, first defie ξ > 0 by ξ /(+) [λv (K) ρ(k, u) + + µv (L) ρ(l, u) + ] /(+) ds(u). (0) S The body λk ˆ+µL ϕ is defied as the body whose radial fuctio is give by ξ ρ(λk ˆ+µL, ) + λv (K) ρ(k, ) + + µv (L) ρ(l, ) +. () It follows immediately that ξ V (λk ˆ+µL), ad hece V (λk ˆ+µL) ρ(λk ˆ+µL, ) + λv (K) ρ(k, ) + + µv (L) ρ(l, ) +. Lutwak [0] defie a mappig: Λ : ϕ c κ c ad poit out that Λ tasforms harmoic Blaschke liear combiatio ito Blaschke liear combiatios, i.e. If K, L ϕ c ad λ, µ 0, the Λ(λK ˆ+µL) λ ΛK +µ ΛL. Further, We obtai that If K j ϕ c (j,..., m), ad λ j 0(j,..., m), the Λ(λ K ˆ+ ˆ+λ m K m ) λ ΛK + +λ m ΛK m. (2) ad Λ(λK) λλk (3) 6

5 ON ANALOGS OF THE DUAL BRUNN-MINKOWSKI INEQUALITY.4 Projectio bodies ad Cetroid bodies The projectio bodies, ΠK, of the body K K is defied as the covex figure whose support fuctio is give, for u S, by [6] h(πk, u) v(k u ) (4) It is easy to see, that a projectio body is always cetered(symmetric about the origi), ad if K has iterior poits the ΠK will have iterior poit as well. Here, we itroduce the followig property. If K, L K ad λ, µ 0, the [7] Further, we may prove that Π(λ K +µ L) λπk + µπl. If K j K (j,..., m) ad λ j 0(j,..., m), the Π(λ K + +λ m K m ) λ ΠK + + λ m ΠK m. (5) The cetroid body, ΓK, of K ϕ, is the covex body whose support fuctio, at x R, is give by [8]: h(γk, x) V (K) Here, we give the followig property: If K j ϕ (j,..., m), ad λ j 0(j,..., m), the K x y dy. (6) Γ(λ K ˆ+ ˆ+λ m K m ) λ ΓK + + λ m ΓK m. (7) If K ϕ, the from (20) it follows that ΓK is cetered. Please see the ext sectio for above iterrelated otatios, defiitios ad their backgroud material.. Mai results Width-itegrals were first cosidered by Blaschke [,p.85] ad later by Hadwiger [2,p.266]. I [3], Lutwak also itroduced the width-itegral of idex i ad proved some importat results, oe of them is the followig Theorem: 7

6 ZHAO CHANGJIAN, WING-SUM CHEUNG, AND MIHÁLY BENCZE Theorem A. If K, L K ad i <, the B i (K + L/() B i (K/() + B i (L/() (8) with equality if ad oly if K ad L have similar width. Sice iequality () is a ew result similar to the followig Bru-Mikowski iequality for the cross-sectioal measures [2, p.249]. Theorem B. If K, L K ad i <, the W i (K + L/() W i (K/() + W i (L/() (9) with equality if ad oly if K ad L are homothetic. Hece, iequality () is called as the dual Bru-Mikowski iequality for width-itegral of covex bodies. The mai purpose of this paper is to establish two aalogs of iequality (), them ca be stated as: Theorem C. If K,..., K m ϕ, λ,..., λ m > 0 ad i <, the B i (Γ(λ K ˆ+ ˆ+λ m K m )/() λ B i (ΓK )/() + + λ m B i (ΓK m /(), (20) with equality if ad oly if ΓK j (j, 2,..., m) have similar width. Theorem D. If K,..., K m ϕ c, λ,..., λ m > 0 ad i <, the B i (Π(Λ(λ K ˆ+... ˆ+λ m K m ))/() λ B i (Π(ΛK )/() λ m B i (Π(ΛK m )/(), (2) with equality if ad oly if Π(ΛK j )(j, 2,..., m) have similar width. 2. A dual Bru-Mikowski iequality about the width-itegrals of cetroid bodies for the harmoic Blaschke liear combiatio The followig dual Bru-Mikowski iequality about the width-itegrals of cetroid bodies will be proved. Theorem C. If K,..., K m ϕ, λ,..., λ m > 0 ad i <, the B i (Γ(λ K ˆ+ ˆ+λ m K m )/() λ B i (ΓK )/() + + λ m B i (ΓK m /(), (22) 8

7 ON ANALOGS OF THE DUAL BRUNN-MINKOWSKI INEQUALITY with equality if ad oly if ΓK j (j, 2,..., m) have similar width. Proof. From (7),(0),(),(2) ad i view of Mikowski iequality for itegral [, p.47], we obtai that ( B i (Γ(λ K ˆ+ ˆ+λ m K m ) b S Γ(λ K ˆ+ ˆ+λ ds(u) mk m) ( λ ( ( S (b λγk + +λ mγk m ) ds(u) S (λ b ΓK + + λ m b ΓKm ) ds(u) S b ΓK ds(u) + + λm ( λ B i (ΓK + + λm B i (ΓK m, S b ΓK m ds(u) with equality if ad oly if ΓK j (j,..., m) have similar width. The proof is complete. Takig m 2 to (23), we have Corollary. If K, L ϕ, λ, µ > 0 ad i <, the B i (Γ(λK ˆ+µL)/() λb i (ΓK/() + µb i (ΓL/(), (23) with equality if ad oly if ΓK ad ΓL have similar width. Aother importat cosequece is obtaied whe λ µ. Corollary 2. IfK, L ϕ ad i <, the B i (Γ(K ˆ+L)/() B i (ΓK/() + B i (ΓL/(), (24) with equality if ad oly if ΓK ad ΓL have similar width. 3. A dual Bru-Mikowski iequality about the width-itegrals of projectio bodies for the harmoic Blaschke liear combiatio The followig dual Bru-Mikowski iequality about the width-itegrals of projectio bodies will be proved. Theorem D. If K,..., K m ϕ c, λ,..., λ m > 0 ad i <, the B i (Π(Λ(λ K ˆ+... ˆ+λ m K m ))/() 9

8 ZHAO CHANGJIAN, WING-SUM CHEUNG, AND MIHÁLY BENCZE λ B i (Π(ΛK )/() λ m B i (Π(ΛK m )/(), (25) with equality if ad oly if Π(ΛK j )(j, 2,..., m) have similar width. Proof. From (7),(0),(),(6),(9) ad i view of Mikowski iequality for itegral [, p.47], we obtai that ( B i (Π(Λ(λ K ˆ+ ˆ+λ m K m )) b S Π(Λ(K ˆ+ ˆ+K ds(u) m)) ( b )ds(u) S Π(λ ΛK + +λ mλk m ( S b m λjπ(λkj)ds(u) S j m λ j b Π(ΛKj) j m ( λ j b Π(ΛK ds(u) j) S j m j ds(u) λ j B i (Π(ΛK j ), with equality if ad oly if Π(ΛK j )(j,..., m) have similar width. The proof is complete. Takig m 2 to (26), we have Corollary 3. If K, L ϕ c, λ, µ > 0 ad i <, the B i (Π(Λ(λK ˆ+µL))/() λb i (Π(ΛK)/() + µb i (Π(ΛL)/(), (26) with equality if ad oly if Π(ΛK) ad Π(ΛL) have similar width. Aother remarkable case is obtaied for λ µ. Corollary 4. If K, L ϕ c ad i <, the B i (Π(Λ(K ˆ+L))/() B i (Π(ΛK)/() + B i (Π(ΛL)/(), (27) with equality if ad oly if Π(ΛK) ad Π(ΛL) have similar width. Ackowledgemets. This work was supported by the Natioal Natural Sciece Foudatio of Chia (Grat No ) ad Academic Maistay of Middleage ad Youth Foudatio of Shadog Provice. 20

9 ON ANALOGS OF THE DUAL BRUNN-MINKOWSKI INEQUALITY Refereces [] Blaschke W., Vorlesugeüber itegralgeometrie, I, II, Teuber, Leipzig, 936, 937; reprit, Chelsea, New York, 949. [2] Boese T., ad Fechel W., Theorie der Kovexe Körper, Spriger, Berli, 934. [3] Hadwiger H., Vorlesugeüber ihalt, Oberfläche ud isoperimetrie, Spriger, Berli, 957. [4] Hardy G. H., Littlewood J. E., ad Pölya G., Iequalities, Cambridge Uiversity Press, Cambridge, 934. [5] Lutwak E., Width-itegrals of covex bodies, Proc. Amer. Math. Soc., 53(975), [6] Lutwak E., Iequalities for mixed projectio, Tas. Amer. Math. Soc., 339(993), [7] Lutwak E., Itersectio bodies ad dual mixed volumes, Adv. Math., 7(988), [8] Lutwak E., Cetroid bodies ad dual mixed volumes, Proc. Lodo Math. Soc., 60(990), [9] Lutwak E., Mixed projectio iequalities, Tras. Amer. Math. Soc., 287(985), [0] Lutwak E., Exteded affie surface area, Adv. Math., 85(99), [] Scheider R., Covex Bodies: The Bru-Mikowski Theory, Cambridge Uiversity Press, Cambridge, 993. Departmet of Mathematics, Shaghai Uiversity, Shaghai, , P. R. Chia, Departmet of Mathematics, Bizhou Teachers College, Shadog, , P. R. Chia address: Departmet of Mathematics, The Uiversity of Hog Kog, Pokfulam Road, Hog Kog Str. Harmaului 6, RO-222 Sacele, Jud. Brasov, Romaia 2