On the Orlicz-Brunn-Minkowski theory

Transcription

1 On the Orlicz-Brunn-Minkowski theory C J Zhao Abstract Recently, Garner, Hug an Weil evelope an Orlicz-Brunn- Minkowski theory Following this, in the paper we further consier the Orlicz-Brunn-Minkowski theory The funamental notions of mixe quermassintegrals, mixe p-quermassintegrals an inequalities are extene to an Orlicz setting Inequalities of Orlicz Minkowski an Brunn-Minkowski type for Orlicz mixe quermassintegrals are obtaine One of these has connections with the conjecture log-brunn-minkowski inequality an we prove a new log-minkowski-type inequality A new version of Orlicz Minkowski s inequality is prove Finally, we show Simon s characterization of relative spheres for the Orlicz mixe quermassintegrals MSC 200: 52A20, 52A30 Key wors: L p aition; Orlicz aition; Orlicz mixe volume; mixe quermassintegrals; mixe p-quermassintegrals, Orlicz mixe quermassintegrals; Orlicz-Minkowski inequality; Orlicz-Brunn-Minkowski inequality Introuction One of the most important operations in geometry is vector aition As an operation between sets K an L, efine by K + L = {x + y x K, y L}, it is usually calle Minkowski aition an combine volume play an important role in the Brunn-Minkowski theory During the last few ecaes, the theory has been extene to L p -Brunn-Minkowski theory The first, a set calle as L p aition, introuce by Firey in [6] an [7] Denote by + p, for p, efine by hk + p L, x p = hk, x p + hl, x p, for all x R n an compact convex sets K an L in R n containing the origin When p =, is interprete as hk + L, x = max{hk, x, hl, x}, as is customary Here the functions are the support functions If K is a nonempty close not necessarily boune convex set in R n, then hk, x = max{x y y K}, Balkan Journal of Geometry an Its Applications, Vol22, No, 207, pp 98-2 c Balkan Society of Geometers, Geometry Balkan Press 207

2 On the Orlicz-Brunn-Minkowski theory for x R n, efines the support function hk, x of K A nonempty close convex set is uniquely etermine by its support function L p aition an inequalities are the funamental an core content in the L p Brunn-Minkowski theory For recent important results an more information from this theory, we refer to [2], [3], [4], [5], [20], [22], [23], [24], [25], [26], [27], [30], [3], [35], [36], [37] an the references therein In recent years, a new extension of L p -Brunn-Minkowski theory is to Orlicz- Brunn-Minkowski theory, initiate by Lutwak, Yang, an Zhang [28] an [29] In these papers the notions of L p -centroi boy an L p -projection boy were extene to an Orlicz setting The Orlicz centroi inequality for star boies was introuce in [39] which is an extension from convex to star boies The other articles avance the theory can be foun in literatures [], [7], [8] an [32] Very recently, Garner, Hug an Weil [9] constructe a general framework for the Orlicz-Brunn-Minkowski theory, an mae clear for the first time the relation to Orlicz spaces an norms They introuce the Orlicz aition K + φ L of compact convex sets K an L in R n containing the origin, implicitly, by hk, x 2 φ hk + φ L, x, hl, x =, hk + φ L, x for x R n, if hk, x+hl, x > 0, an by hk + φ L, x = 0, if hk, x = hl, x = 0 Here φ Φ 2, the set of convex functions φ : [0, 2 [0, that are increasing in each variable an satisfy φ0, 0 = 0 an φ, 0 = φ0, = Unlike the L p case, an Orlicz scalar multiplication cannot generally be consiere separately The particular instance of interest correspons to using 2 with φx, x 2 = φ x + εφ 2 x 2 for ε > 0 an some φ, φ 2 Φ, in which case we write K + φ,ε L instea of K + φ L, where the sets of convex function φ i : [0, 0, that are increasing an satisfy φ i = an φ i 0 = 0, where i =, 2 Orlicz aition reuces to L p aition, p <, when φx, x 2 = x p + xp 2, or L aition, when φx, x 2 = max{x, x 2 } Moreover, Garner, Hug an Weil [9] introuce the Orlicz mixe volume, obtaining the equation φ 3 l V K + φ,ε L V K = hl, u φ 2 hk, usk, u, n ε 0 + ε n S n hk, u where SK, u is the mixe surface area measure of K an φ Φ 2, φ, φ 2 Φ Here K is a convex boy containing the origin in its interior an L is a compact convex set containing the origin, assumptions we shall retain for the remainer of this introuction Denoting by V φ K, L, for any φ Φ, the integral on the right sie of 3 with φ 2 replace by φ, we see that either sie of the equation 3 is equal to V φ2 K, L an therefore this new Orlicz mixe volume plays the same role as V p K, L in the L p -Brunn-Minkowski theory In [9], Garner, Hug an Weil obtaine the Orlicz- Minkowksi inequality /n V L 4 V φ K, L V K φ, V K for φ Φ If φ is strictly convex, equality hols if an only if K an L are ilates or L = {o}

3 00 C J Zhao In Section 3, we compute the Orlicz first variation of quermassintegrals, call as Orlicz mixe quermassintegrals, obtaining the equation 5 φ l W i K + φ,ε L W i K = hl, u φ 2 hk, us i K, u n i ε 0 + ε n S hk, u n for φ Φ 2, φ, φ 2 Φ an i n, an W i enotes the usual quermassintegrals, an S i K, u is the i-th mixe surface area measure of K Denoting by W φ,i K, L, for any φ Φ, the integral on the right sie of 5 with φ 2 replace by φ, we see that either sie of the equation 5 is equal to W φ2,ik, L an therefore this new Orlicz mixe volume Orlicz mixe quermassintegrals plays the same role as W p,i K, L in the L p -Brunn-Minkowski theory Note that when i = 0, 5 becomes 3 Hence we have the following efinition of Orlicz mixe quermassintegrals 6 W φ,i K, L = hl, u hk, us i K, u n hk, u S n φ In Section 4, we establish Orlicz-Minkowksi inequality for the Orlicz mixe quermassintegrals /n i Wi L 7 W φ,i K, L W i K φ, W i K for φ Φ an 0 i < n If φ is strictly convex, equality hols if an only if K an L are ilates or L = {o} Note that when i = 0, 7 becomes to 4 In particularly, putting φt = t p, p < in 7, 7 reuces to the following L p -Minkowski inequality for mixe p-quermassintegrals establishe by Lutwak [2] 8 W p,i K, L n i W i K n i p W i L p, for p > an 0 i n, with equality if an only if K an L are ilates or L = {o} Putting i = 0, φt = t p an p < in 7, 7 reuces to the well-known L p -Minkowski inequality establishe by Firey [7] For p >, 9 V p K, L V K n p/n V L p/n, with equality if an only if K an L are ilates or L = {o} In Section 5, we establish the following Orlicz-Brunn-Minkowksi inequality for quermassintegrals of Orlicz aition /n i /n i Wi K W i L 0 φ,, W i K + φ L W i K + φ L for φ Φ 2 an 0 i < n If φ is strictly convex, equality hols if an only if K an L are ilates or L = {o} Note that when φx, x 2 = x p + xp 2, p < in, 80 reuces to the following L p -Brunn-Minkowski inequality for quermassintegrals 8 establishe by Lutwak [2] If W i K + p L p/n i W i K p/n i + W i L p/n i,

4 On the Orlicz-Brunn-Minkowski theory 0 82 with equality if an only if K an L are ilates or L = {o}, an where p an 0 i < n Putting i = 0, φx, x 2 = x p +xp 2 an p < in, reuces to the well-known L p -Brunn-Minkowski inequality establishe by Firey [7] 2 V K + p L p/n V K p/n + V L p/n, with equality if an only if K an L are ilates or L = {o}, an where p > A special case of 0 was recently establishe by Garner, Hug an Weil [9] /n /n V K V L 3 φ,, V K + φ,ε L V K + φ L for φ Φ 2 If φ is strictly convex, equality hols if an only if K an L are ilates or L = {o} When i = 0, 0 becomes to 2 Moreover, We prove also the Orlicz Minkowski inequality 4 an the Orlicz Brunn-Minkowski inequality 2 are equivalent, an 7 an 0 also are equivalent When we were about to submit our paper, we were informe that G Xiong an D Zou [38] ha also obtaine Orlicz Minowski an Brunn-Mingkowski inequalities for Orlicz mixe quermassintegrals Please note that we use a completely ifferent approach, although the two inequalities coincie with theirs In 202, Böröczky, Lutwak, Yang, an Zhang [2] conjecture a log-minkowski inequality for origin-symmetric convex boies K an L in R n hl, u V L 4 log hk, usk, u V K log S hk, u V K n In [2], 4 is prove by them only when n = 2 Very recently, Garner, Hug an Weil [9] prove a new version of 4 for convex boies, not origin-symmetric convex boies hl, u 5 log hk, usk, u V K log V n L/n, S hk, u V K n /n with equality if an only if K an L are ilates or L = {o}, an where L intk They also shown that combining 4 an 5 may get the classical Brunn-Minkowski inequality In Section 6, we give a new log-minkowski-type inequality hl, u 6 hk, us i K, u W i K log W il /n i n, hk, u W i K /n i S n log with equality if an only if K an L are ilates or L = {o} When i = 0, 6 becomes 5 We also point out a conjecture which is an extension of the log Minkowski inequality as follows /n i hl, u Wi L 7 log hk, us i K, u log n S hk, u W n i K When i = 0, 7 becomes the log-minkowski inequality 4 Combining 6 an 7 together split the following classical Brunn-Minkowski inequality for quermassintegrals see Section 6 W i K + L /n i W i K /n i + W i L /n i,

5 02 C J Zhao with equality if an only if K an L are ilates or L = {o} In 200, the Orlicz projection boy Π φ of K K is a convex boy containing the origin in its interior efine by Lutwak, Yang an Zhang [28] { } u υ 8 hπ φ, u = inf λ > 0 φ hk, υsk, υ, nv K S λhk, υ n for φ Φ an u S n A ifferent Orlicz version of Minkowski s inequality 8 is presente in Section 7 This results from replacing the left sie of 8 by the quantity 9 { } hl, u Ŵ φ,i K, L = inf λ > 0 φ hk, us i K, u, nw i K S λhk, u n for φ Φ an 0 i < n We prove the following new Orlicz Minkowski type inequality 20 Ŵ φ,i K, L /n i Wi L, W i K where φ Φ an i < n If φ is strictly convex an W i L > 0, equality hols if an only if K an L are ilates A special version of 20 was recently establishe by Garner, Hug an Weil [9] V φ K, L /n V L, V K If φ is strictly convex an V L > 0, then equality hols if an only if K an L are ilates an where { } hl, u V φ K, L = inf λ > 0 φ hk, usk, u, nv K S λhk, u n for φ Φ Finally, in Section 8, we show Simon s characterization of relative spheres for the Orlicz mixe quermassintegrals 2 Notations an preinaries The setting for this paper is n-imensional Eucliean space R n Let K n be the class of nonempty compact convex subsets of R n, let Ko n be the class of members of K n containing the origin, an let Koo n be those sets in K n containing the origin in their interiors A set K K n is calle a convex boy if its interior is nonempty We reserve the letter u S n for unit vectors, an the letter B for the unit ball centere at the origin The surface of B is S n For a compact set K, we write V K for the n-imensional Lebesgue measure of K an call this the volume of K If K is a nonempty close not necessarily boune convex set, then hk, x = sup{x y y K},

6 On the Orlicz-Brunn-Minkowski theory 03 for x R n, efines the support function of K, where x y enotes the usual inner prouct x an y in R n A nonempty close convex set is uniquely etermine by its support function Support function is homogeneous of egree, that is, hk, rx = rhk, x, for all x R n an r 0 Let enote the Hausorff metric on K n, ie, for K, L K n, K, L = hk, u hl, u, where enotes the sup-norm on the space of continuous functions CS n Throughout the paper, the stanar orthonormal basis for R n will be {e,, e n } Let Φ n, n N, enote the set of convex functions φ : [0, n [0, that are strictly increasing in each variable an satisfy φ0 = 0 an φe j = > 0, j =,, n When n =, we shall write Φ instea of Φ The left erivative an right erivative of a real-value function f are enote by f l an f r, respectively 2 Mixe quermassintegrals If K i K n i =, 2,, r an λ i i =, 2,, r are nonnegative real numbers, then of funamental importance is the fact that the volume of r i= λ ik i is a homogeneous polynomial in λ i given by see eg [3] 2 V λ K + + λ n K n = λ i λ in V ii n, i,,i n where the sum is taken over all n-tuples i,, i n of positive integers not exceeing r The coefficient V i i n epens only on the boies K i,, K in an is uniquely etermine by 2, it is calle the mixe volume of K i,, K in, an is written as V K i,, K in Let K = = K n i = K an K n i+ = = K n = L, then the mixe volume V K,, K n is written as V K[n i], L[i] If K = = K n i = K, K n i+ = = K n = B The mixe volumes V i K[n i], B[i] is written as W i K an call as quermassintegrals or i-th mixe quermassintegrals of K We write W i K, L for the mixe volume V K[n i ], B[i], L[] an call as mixe quermassintegrals Aleksanrov [] an Fenchel an Jessen [5] also see Busemann [4] an Schneier [33] have shown that for K Koo, n an i = 0,,, n, there exists a regular Borel measure S i K, on S n, such that the mixe quermassintegrals W i K, L has the following representation: 22 W i K, L = n i W i K + εl W i K = hl, us i K, u ε 0 + ε n S n Associate with K,, K n K n is a Borel measure SK,, K n, on S n, calle the mixe surface area measure of K,, K n, which has the property that for each K K n see eg [8], p353, 23 V K,, K n, K = hk, usk,, K n, u n S n In fact, the measure SK,, K n, can be efine by the propter that 23 hols for all K K n Let K = = K n i = K an K n i = = K n = L, then the mixe surface area measure SK,, K n, is written as SK[n i], L[i],

7 04 C J Zhao When L = B, SK[n i], L[i], is written as S i K, an calle as i-th mixe surface area measure A funamental inequality for mixe quermassintegrals stats that: For K, L K n an 0 i < n, 24 W i K, L n i W i K n i W i L, with equality if an only if K an L are homothetic an L = {o} Goo general references for this material are [4] an [9] 22 Mixe p-quermassintegrals Mixe quermassintegrals are, of course, the first variation of the orinary quermassintegrals, with respect to Minkowski aition The mixe quermassintegrals W p,0 K, L, W p, K, L,, W p,n K, L, as the first variation of the orinary quermassintegrals, with respect to Firey aition: For K, L Koo, n an real p, efine by see eg [2] 25 W p,i K, L = p n i ε 0 + W i K + p ε L W i K ε The mixe p-quermassintegrals W p,i K, L, for all K, L Koo, n has the following integral representation: 26 W p,i K, L = hl, u p S p,i K, u, n S n where S p,i K, enotes the Boel measure on S n The measure S p,i K, is absolutely continuous with respect to S i K,, an has Raon-Nikoym erivative 27 S p,i K, S i K, = hk, p, where S i K, is a regular Boel measure on S n The measure S n K, is inepenent of the boy K, an is just orinary Lebesgue measure, S, on S n S i B, enotes the i-th surface area measure of the unit ball in R n In fact, S i B, = S for all i The surface area measure S 0 K, just is SK, When i = 0, S p,i K, is written as S p K, see [25], [26] A funamental inequality for mixe p-quermassintegrals stats that: For K, L Koo, n p > an 0 i < n, 28 W p,i K, L n i W i K n i p W i L p, with equality if an only if K an L are homothetic L p -Brunn-Minkowski inequality for quermassintegrals establishe by Lutwak [2] If K Koo, n L Ko n an p an 0 i n, then 29 W i K + p L p/n i W i K p/n i + W i L p/n i, with equality if an only if K an L are ilates or L = {o} Obviously, putting i = 0 in 26, the mixe p-quermassintegrals W p,i K, L become the well-known L p -mixe volume V p K, L, efine by see eg [25] 20 V p K, L = hl, u p S p K, u n S n

8 On the Orlicz-Brunn-Minkowski theory The Orlicz mixe volume For φ Φ, K Koo n an L Ko n, Garner, Hug an Weil [9] efine the Orlicz mixe volumes, V φ K, L by 2 V φ K, L = hl, u hk, usk, u n hk, u S n φ They obtaine the Orlicz-Minkowksi inequality /n V L 22 V φ K, L V K φ, V K for all K Koo, n L Ko n an φ Φ If φ is strictly convex, equality hols if an only if K an L are ilates or L = {o} Orlicz mixe quermassintegrals is efine in Section 3, by 23 W φ,i K, L =: hl, u hk, us i K, u, n hk, u S n φ for all K Koo, n L Ko n, φ Φ an 0 i < n Obviously, when φt = t p an p, Orlicz mixe quermassintegrals reuces to the mixe p-quermassintegrals W p,i K, L efine in 26 When i = 0, 23 reuces to 2 24 Orlicz aition Let m 2, φ Φ m, K j K0 n an j =,, m, we efine the Orlicz aition of K,, K m, enote by + φ K,, K m, is efine by { hk, x 24 h+ φ K,, K m, x = inf λ > 0 φ,, hk } m, x, λ λ for x R n Equivalently, the Orlicz aition + φ K,, K m can be efine implicitly an uniquely by hk, x 25 φ h+ φ K,, K m, x,, hk m, x =, h+ φ K,, K m, x for all x R n An important special case is obtaine when m φx,, x m = φ j x j, j= for some fixe φ j Φ such that φ = = φ m = We then write + φ K,, K m = K + φ + φ K m This means that K + φ + φ K m is efine either by m 26 hk + φ + φ K m, u = sup λ > 0 hkj, x φ j λ, j=

9 06 C J Zhao 92 for all x R n, or by the corresponing special case of 25 For real p, K, L Koo n an α, β 0 not both zero, the Firey linear combination α K + p β L Ko n can be efine by see [6] an [7] hα K + p β L, p = αhk, p + βhl, p Obviously, Firey an Minkowski scalar multiplications are relate by α K = α /p K In [9], Garner, Hug an Weil efine the Orlicz linear combination + φ K, L, α, β for K, L Ko n an α, β 0, efine by hk, x hl, x 27 αφ + βφ 2 =, h+ φ K, L, α, β, x h+ φ K, L, α, β, x if αhk, x+βhl, x > 0, an by h+ φ K, L, α, β, x = 0 if αhk, x+βhl, x = 0, for all x R n It is easy to verify that when φ t = φ 2 t = t p, p, the Orlicz linear combination + φ K, L, α, β equals the Firey combination α K + p β L Henceforth we shall write K + φ,ε L instea of + φ K, L,, ε, for ε 0, an assume throughout that this is efine by 27, where α =, β = ε, an φ, φ 2 Φ 3 Orlicz mixe quermassintegrals In orer to efine a new concept: Orlicz mixe quermassintegrals, we nee Lemmas 3-34 an Theorem 35 Lemma 3 [9] If φ Φ m, then Orlicz aition + φ : K0 n m K0 n is continuous, GLn covariant, monotonic, projection covariant an has the ientity property Lemma 32 [9] If K, L K n o, then 3 K + φ,ε L K, in the Hausorff metric as ε 0 + Lemma 33 If K, L Ko n an 0 i < n, Then 32 W i K + φ,ε L W i K = n i hk + φ,ε L, u hk, u S i K, u, ε 0 + ε n ε 0 + ε where, ε 0 + S n hk+ φ,εl,u hk,u ε uniformly for u S n Proof For brevity, we temporarily write K ε = K + φ,ε L Starting with the ecomposition W i K ε W i K ε = n i j=0 W i K ε [j + ], K[n i j ] W i K ε [j], K[n i j] ε 20 Notice that 33 W i K ε [j + ], K[n i j ] W i K ε [j], K[n i j] ε

10 On the Orlicz-Brunn-Minkowski theory 07 = hk ε, u hk, u S i K ε [j], K[n i j ], u n S ε n = hkε, u hk, u hk + φ,ε L, u hk, u n ε ε 0 + ε + n S n S n S i K ε [j], K[n i j ], u hk + φ,ε L, u hk, u S i K ε [j], K[n i j ], u ε 0 + ε By assumption, the integran in 33 converges uniformly to zero for u S n Since K ε K as ε 0 +, by Lemma 32, an the i-th mixe surface area measures S i K ε [j], K[n i j ] are uniformly boune for ε 0, ], the first integral in the previous sum converges to zero Noting that S i K ε [j], K[n i j ] S i K, u weakly as ε 0 + Hence n i W i K + φ,ε L W i K = ε 0 + ε ε 0 + n j=0 S n hk + φ,ε L, u hk, u ε 0 + ε S i K ε [j], K[n i j ], u = n i hk + φ,ε L, u hk, u S i K, u n ε 0 + ε S n Lemma 34 For ε > 0 an u S n, let h ε = hk + φ,ε L, u If K Koo n an L Ko n, then 34 h ε ε = hl, u φ εφ 2 h ε hk, u φ y hl, u φ y 2 h ε 2 + ε hl, uhl n, u h 2 ε φ y y, φ 2 z z where an hl, u y = εφ 2, h ε z = hl, u h ε Proof Suppose ε > 0, L K n o, K K n oo an u S n, an notice that h ε = hk + φ,ε L, u, we have hk, u h ε hl, u = φ εφ 2 h ε

11 08 C J Zhao 24 On the other han where an h ε ε = ε = hk, u φ y y hk, u hl, u φ εφ 2 h ε [ hl, u φ 2 h ε φ εφ 2 hl, u h ε hl, u y = εφ 2, h ε z = hl, u h ε By simplifying the equation above, it easy follows 34 ε φ ] 2z hl, u h ε z h 2 ε ε Theorem 35 Let φ Φ 2, an φ, φ 2 Φ If K Koo, n L Ko n an i n, then 35 φ l W i K + φ,ε L W i K = hl, u φ 2 hk, us i K, u n i ε 0 + ε n S hk, u n Proof From Lemma 33, we obtain W i K + φ,ε L W i K = n i hk + φ,ε L, u hk, u S i K, u ε 0 + ε n S n ε 0 + ε = n i h ε n ε 0 + ε S ik; u S n From Lemmas 3-32 an Lemma 34, an noting that y as ε 0 +, we have φ y φ y φ = = ε y y φ l, 27 the equation 35 yiels easy The theorem plays a central role in our eriving new concept of the Orlicz mixe quermassintegrals Here, we give the another proof Proof From the hypotheses, we have for ε > 0 hk + φ,ε L, u = φ hk, u εφ 2 hl, u hk + φ,ε L, u

12 On the Orlicz-Brunn-Minkowski theory Hence ε 0 + hk + φ,ε L, u hk, u ε where = ε 0 + = ε 0 + φ φ hk, u hk, u hl, u εφ 2 hk + φ,ε L, u ε hl, u hk, uφ 2 hk + φ,ε L, u φ 2 hl, u y εφ 2 hk + φ,ε L, u hl, u y = εφ 2, hk + φ,ε L, u an note that y as ε o + Notice that φ y y φ = y φ l, an from 22,36 an Lemmas 3-32, 35 easy follows y φ, y Denoting by W φ,i K, L, for any φ Φ an i < n, the integral on the righthan sie of 35 with φ 2 replace by φ, we see that either sie of the equation 35 is equal to W φ2,ik, L an therefore this new Orlicz mixe volume W φ,i K, L Orlicz mixe quermassintegrals has been born Definition 3 Orlicz mixe quermassintegrals For φ Φ, Orlicz mixe quermassintegrals, W φ,i K, L, for 0 i < n, efine by 37 W φ,i K, L =: n for all K K n oo, L K n o S n φ hl, u hk, us i K, u, hk, u Remark 32 Let φ t = φ 2 t = t p, p in 35, the Orlicz sum K + φ,ε L reuces to the L p aition K + p ε L, an the Orlicz mixe quermassintegrals W φ,i K, L become the well-known mixe p-quermassintegrals W p,i K, L Obviously, when i = 0, W φ,i K, L reuces to Orlicz mixe volumes V φ K, L efine by Garner, Hug an Weil [9] Theorem 36 If φ, φ 2 Φ, φ Φ 2 an K K n o, L K n oo, an 0 i < n, then W φ2,ik, L = φ l n i W i K + φ,ε L W i K ε 0 + ε Proof This follows immeiately from Theorem 35 an 37

13 0 C J Zhao Orlicz-Minkowski type inequality In the Section, we nee efine a Borel measure in S n, Wn,i K, υ, calle as i-th normalize cone measure Definition 4 If K Koo, n i-th normalize cone measure, Wn,i K, υ, efine by 4 W n,i K, υ = hk, υ nw i K S ik, υ When i = 0, W n,i K, υ becomes to the well-known normalize cone measure V n K, υ, by 42 V n K, υ = hk, υ SK, υ nv K This was efine in [2] an [9] In the following, we start with two auxiliary results Lemmas 4 an 42, which will be the base of our further stuy The Orlicz-Minkowski inequality for Orlicz mixe quermassintegrals is establishe in Theorem 43 Lemma 4 Jensen s inequality Suppose that µ is a probability measure on a space X an g : X I R is a µ-integrable function, where I is a possibly infinite interval If φ : I R is a convex function, then 43 φgxµx φ gxµx X If φ is strictly convex, equality hols if an only if gx is constant for µ-almost all x X see [6] Lemma 42 Let 0 < a be an extene real number, an let I = [0, a be a possibly infinite interval Suppose that φ : I [0, is convex with φ0 = 0 If K Koo n an L Ko n are such that L intak, then /n i hl, u Wi L 44 φ hk, us i K, u φ nw i K S hk, u W n i K If φ is strictly convex, equality hols if an only if K an L are ilates or L = {o} X Proof In view of L intak, so 0 hl,u hk,u < a for all u Sn By 4 an note that 22 with K = L, it follows the i-th normalize cone measure W n,i K, u is a probability measure on S n Hence by using Jensen s inequality 43, the Minkowski s inequality 24, an the fact that φ is increasing, to obtain nw i K S n φ hl, u hk, us i K, u = hk, u φ S n φ Wi K, L W i K hl, u hk, u W n,i K, u

14 On the Orlicz-Brunn-Minkowski theory φ /n i Wi L W i K In the following, we iscuss the equal conition of 44 Suppose the equality hols in 44 an φ is strictly convex, so that φ > 0 on 0, a Moreover, notice the injectivity of φ, we have equality in Minkowski inequality 24, so there are r 0 an x R n such that L = rk + x an hence hl, u = rhk, u + x u for all u S n Since equality must hol in Jensen s inequality 43 as well, when φ is strictly convex we can conclue from the equality conition for Jensen s inequality that hl, u 46 nw i K S hk, u hk, us hl, v ik, u = hk, v, n for S i K, -almost all v S n Hence r + nw i K S n x u hk, us i K, u = r + hk, u x v hk, v, for S i K, -almost all v S n From this an the fact that the centroi of S i K, is at the origin, we get 0 = x us i K, u = nw i K S n x us i K, u = nw i K S n x v hk, v, that is, x v = 0, for S i K, -almost all v S n Hence x = o, namely L = rk Theorem 43 Let φ Φ If K K n oo, L K n o an 0 i < n, then 47 W φ,i K, L W i K φ /n i Wi L W i K 26 If φ is strictly convex, equality hols if an only if K an L are ilates or L = {o} Proof This follows immeiately from 37 an Lemma 42, with a = Corollary 44 [2] If K Koo n an L Ko n, an p > an 0 i n, then W p,i K, L n i W i K n i p W i L p, with equality if an only if K an L are ilates or L = {o} Proof This follows immeiately from 47 with φt = t p an p >

15 2 C J Zhao Remark 42 When a =, putting φt = e t in 44, we obtain 48 log exp S n hl, u hk, u W n,i K, u Similarly, L p -Minkowski inequality 8 can be written as 49 S n p /p hl, u hk, u W n,ik, u /n i Wi L W i K /n i Wi L W i K When p =, 49 becomes to a new form of the Minkowski inequality 24 The left sie of 49 is just the pth mean of the function hl, u/hk, u with respect to W n,i K, Notice that pth means increase with p >, so we fin that the Minkowski inequality 24 implies L p -Minkowski inequality 28 5 Orlicz-Brunn-Minkowski type inequality In this section, we establish the Orlicz Brunn-Minkowski inequality for Orlicz mixe quermassintegrals Theorem 5 Let φ Φ 2 If K Koo, n L Ko n an i < n, then Wi K /n i 5 φ W i K + φ L, W i L /n i /n i W i K + φ L /n i If φ is strictly convex, equality hols if an only if K an L are ilates or L = {o} Proof From the hypotheses an Theorem 43, we obtain 52 W i K + φ L = φ n S n = hk, u φ n hk + φ L, u S n hk, u hk + φ L, u, hl, u hk + φ L, u + φ 2 = W φ,ik + φ L, K + W φ2,ik + φ L, L Wi K /n i W i K + φ Lφ W i K + φ L, /n i hk + φ L, us i K + φ L, u hl, u hk + φ L, u W i L /n i W i K + φ L /n i hk + φ L, us i K + φ L, u This is just 5 If equality hols in 52, then in 52, with K, L an φ replace by K + φ L, K an φ an by K + φ L, L an φ 2, respectively So if φ is strictly convex, then φ an φ 2 are also, so both K an L are multiples of K + φ L, an hence are ilates of each other or L = {o} Corollary 52 [2] If p >, K K n oo, L K n o, while 0 i < n, then 53 W i K + p L p/n i W i K p/n i + W i L p/n i, 283 with equality if an only if K an L are ilates or L = {o}

16 On the Orlicz-Brunn-Minkowski theory Proof The result follows immeiately from Theorem 5 with φx, x 2 = x p + xp 2 an p > Theorem 53 Orlicz Brunn-Minkowski inequality for Orlicz mixe quermassintegrals implies Orlicz Minkowski inequality for Orlicz mixe quermassintegrals Proof Since φ is increasing, so φ is also increasing an hence from 5, we obtain for ε > 0 W i K + φ,ε L φ W i K /n i n i W εφ i L 2 W i K + φ,ε L From Theorem 36, we obtain ε 0 + where φ = φ l ε 0 + W φ2,ik, L φ l n i W i K /n i n i W i K W i L εφ 2 W i K + φ,ε L φ φ ε W i K /n i 2n i W i L εφ 2 W i K + φ,ε L εφ 2 W i L W i K + φ,ε L /n i W i L φ 2 W i K + φ,ε L /n i n i φ z /n i W i L z = εφ 2, W i K + φ,ε L z φ, z an note that z as ε o + On the other han, in view of φ z 0 + z φ = z φ l, an from Lemma 32 Hence /n i Wi L 54 W φ2,ik, L W i Kφ 2 W i K Replace φ 2 by φ, this yiels the Orlicz Minkowski inequality in 47 The equality conition follows immeiately from the equality of Orlicz Brunn-Minkowski inequality for Orlicz mixe quermassintegrals

17 4 C J Zhao From the proof of Theorem 5, we may see that Orlicz Minkowski inequality for Orlicz mixe quermassintegrals implies also Orlicz Brunn-Minkowski inequality for Orlicz mixe quermassintegrals, an this combines Theorem 53, we foun that Theorem 54 Orlicz Brunn-Minkowski inequality for Orlicz mixe quermassintegrals is equivalent to Orlicz Minkowski inequality for Orlicz mixe quermassintegrals Namely: Let φ 2 Φ an φ Φ 2 If K Koo, n L Ko n an i < n, then /n i Wi L 55 W φ2,ik, L W i Kφ 2 W i K Wi K /n i φ W i K + φ L, W i L /n i /n i W i K + φ L /n i If φ is strictly convex, equality hols if an only if K an L are ilates or L = {o} Corollary 55 Orlicz ual Brunn-Minkowski inequality is equivalent to Orlicz ual Minkowski inequality Namely: Let φ 2 Φ an φ Φ 2 If K Koo n an L Ko n, then 56 /n V L V K /n V φ2 K, L V Kφ 2 φ V K V K + φ L, V L /n /n V K + φ L /n If φ is strictly convex, equality hols if an only if K an L are ilates or L = {o} Proof The result follows immeiately from Theorem 54 with i = 0 6 The log-minkowski type inequality Assume that K, L K n oo, then the log Minkowski combination, λ K + o λ L, is efine by λ K + o λ L = u S n {x R n x u hk, u λ hl, u λ }, for all real λ [0, ] Böröczky, Lutwak, Yang, an Zhang [2] conjecture that for origin-symmetric convex boies K an L in R n an 0 λ, 6 V λ K + o λ L V K λ V L λ In [2], they prove 6 only when n = 2 an K, L are origin-symmetric convex boies, an note that while it is not true for general convex boies Moreover, they also shown that 6, for all n, is equivalent to the following log-minkowski inequality hl, u 62 log S hk, u V n K, υ V L n log, V K n where V n K, is the normalize cone measure for K In fact, replacing K an L by K + L an K, respectively, 62 becomes to the following /n hk, u 63 log S hk + L, u V V K n K + L, u log V K + L n

18 On the Orlicz-Brunn-Minkowski theory In [9], Garner, Hug an Weil gave a new version of 63 for the nonempty compact convex subsets K an L, not origin-symmetric convex boies, as follows If K Koo n an L Ko n, then 64 S n log hk, u hk + L, u V n K + L, u log V K + L /n V L /n V K + L /n, with equality if an only if K an L are ilates or L = {o} They also shown that combining 63 an 64, may get the classical Brunn-Minkowski inequality V K + L /n V L /n V K /n, whenever K Koo n an L Ko n an 62 hols with K an L replace by K + L an K, respectively In particular, if 62 hols as it oes, for origin-symmetric convex boies when n = 2, then 62 an 64 together split the classical Brunn-Minkowski inequality In the following, we give a new version of 64 Lemma 6 If K Koo n an L Ko n are such that L intk an i < n, then 65 Wi K /n i W i L /n i hk, u hl, u log log W i K /n i S hk, u W n,i K, u, n with equality if an only if K an L are ilates or L = {o} Proof Since K Koo n an L Ko n are such that L intk Let φt = log t, an notice that φ0 = 0 an φ is strictly increasing an strictly convex on [0, with φt as t Hence the inequality 65 is a irect consequence of Lemma 43 with this choice of φ an a = Theorem 62 If K Koo, n L Ko n an i < n, then 66 Wi K + L /n i W i L /n i hk, u log log W i K + L /n i S hk + L, u W n,i K+L, u, n with equality if an only if K an L are ilates or L = {o} Proof If K Koo n an L Ko n, then K + L Koo n In view of L intk + L an from Lemma 6 with K replace by K + L, 66 easy follows Putting i = 0 in 66, 66 reuces to 64 Here, we point out a new conjecture which is an extension of the log Minkowski inequality 62: Conjecture If K Koo, n L Ko n an i < n, then hl, u 67 log S hk, u W n,i K, u n i log Wi L W n i K Corollary 63 If K Koo, n L Ko n an i < n, then hk, u 68 log S hk + L, u W n,i K + L, u n i log Wi K W n i K + L

19 6 C J Zhao Proof The result follows immeiately from 67 with replacing K an L by K + L an K, respectively It is easy that combine 66 an 68 together split the following classical Brunn- Minkowski inequality for quermassintegrals If K K n oo, L K n o an 0 i n, then W i K + L /n i W i K /n i + W i L /n i, 335 with equality if an only if K an L are ilates or L = {o} A new version of Orlicz Minkowski s inequality In 200, the Orlicz projection boy Π φ of K efine by Lutwak, Yang an Zhang [28] { } u υ 7 hπ φ, u = inf λ > 0 φ S λhk, υ V n K, υ, n for K Koo, n u S n, where V n K, is the normalize cone measure for K Here, we efine the i-th Orlicz mixe projection boy Definition 7 Let K Koo, n L Ko n, φ Φ an 0 i < n, the i-th Orlicz mixe projection boy, Π φ,i, efine by { } u υ 72 hπ φ,i, u = inf λ > 0 φ S λhk, υ W n,i K, υ, n for u S n, where W n,i K, is the i-th normalize cone measure for K efine in 4 Obviously, when i = 0, 72 becomes 7 In the Section, efinition 7 of the i-th Orlicz projection boy suggests efining, by analogy, { } hl, u 73 Ŵ φ,i K, L = inf λ > 0 φ S λhk, u W n,i K, u, n an call as Ŵφ,iK, L Orlicz type quermassintegrals Theorem 7 If φ Φ an K K n oo, L K n o an i < n, then 74 Ŵ φ,i K, L /n i Wi L W i K If φ is strictly convex an W i L > 0, equality hols if an only if K an L are ilates Proof Replacing K by λk, λ > 0 in 44 with a =, we have 75 S n φ hl, u λhk, u W n,i K, u φ λ /n i Wi L W i K

20 On the Orlicz-Brunn-Minkowski theory 7 35 Let Hence S n φ φ λ In view of φ is strictly increasing, we obtain hl, u λhk, u W n,i K, u /n i Wi L W i K 76 /n i Wi L λ W i K From 73 an 76, 74 easy follows In the following, we iscuss the equality conition of 74 Suppose that equality hols, φ is strictly convex an W i L > 0 From 73, the exist µ = Ŵφ,iK, L > 0 satisfies hl, u µhk, u W n,i K, υ = Hence namely: S n φ φ µ = µ /n i Wi L ; W i K /n i Wi L = W i K Therefore the equality in 75 hols for λ = µ From the equality conition of 44, it follows µk an L are ilates When φt = t p an p in 73, it easy follows that Ŵ φ,i K, L = /p Wp,i K, L W i K Putting φt = t p an p in 74, 74 reuces to the classical L p -Minkowski inequality 8 for mixe p-quermassintegrals There is no irect relationship between the Orlicz-Minkowski inequalities 47 an 74 Inee, when φ > 0 on 0,, these can be written in the forms /n i W φ,i K, L Wi L φ, 77 W i K W i K an /n i Wi L 77 φ Ŵφ,i K, L φ W i K respectively, an each of the two quantities on the left-han sies can be larger than the other This is very interesting

21 8 C J Zhao Simon s characterization of relative spheres Theorem 8 Suppose K Koo, n L Ko n, an S Ko n is a class of boies such that K, L S If 0 i < n an φ Φ, an 8 W φ,i Q, K = W φ,i Q, L, for all Q S, 363 then K = L Proof To see this take Q = K, an from 30 an Theorem 44, we have /n i Wi L W i K = W φ,i K, K = W φ,i K, L W i Kφ W i K If φ is strictly convex, equality hols if an only if K an L are ilates or L = {o} Hence /n i Wi L φ W i K If φ is strictly convex, equality hols if an only if K an L are ilates or L = {o} Note that φ is increasing, we obtain Take Q = L, we have W i L W i K W i L = W φ,i L, L = W φ,i L, K W i Lφ Wi /n i K W i L If φ is strictly convex, equality hols if an only if K an L are ilates or L = {o} Hence Wi /n i K φ W i L 364 If φ is strictly convex, equality hols if an only if K an L are ilates or L = {o} Hence W i K W i L This yiels W i K = W i L Hence K = L Corollary 82 Suppose K Koo, n L Ko n, an S Ko n that K, L S If φ Φ, an is a class of boies such 82 V φ Q, K = V φ Q, L, for all Q S, then K = L Proof The result follows immeiately from Theorem 8 with i = Putting φt = t p an p > in Theorem 8, we obtain the following result which was prove by Lutwak [2]

22 On the Orlicz-Brunn-Minkowski theory Corollary 83 Suppose K Koo, n L Ko n, an S Ko n is a class of boies such that K, L S If p >, 0 i < n, an 83 W p,i Q, K = W p,i Q, L, for all Q S, then K = L Theorem 84 Suppose 0 i < n an φ Φ For K Koo, n the following statements are equivalent: i The boy K is centere, ii The measure W n,i K, is even iii W φ,i K, Q = W φ,i K, Q, for all Q Koo n iv W φ,i K, Q = W φ,i K, Q, for Q = K Proof To see that i implies ii, recall that if K is centere, then hk, is an even function, an S i K is an even measure The implication is now a consequence of the fact that W n,i K, = nw i K hk, S ik, That ii yiels iii is a consequence of the following integra representation hq, u W φ,i K, Q = W i K hk, u W n,i K, u, S n φ an the fact that, in general, h Q, u = hq, u, for all u S n Obviously, iv follows irectly from iii To see that iv implies i, notice that iv, for Q = K, gives W i K = W φ,i K, K The esire result follows from the fact that W i K = W i K an the equality conitions of the Orlicz-Minkoski inequality 47 Corollary 85 Suppose φ Φ For K Koo, n the following statements are equivalent: i The boy K is centere, ii The measure V n K, is even iii V φ K, Q = V φ K, Q, for all Q Koo n iv V φ K, Q = V φ,i K, Q, for Q = K Proof The results follow immeiately from Theorem 85 with i = 0 Corollary 86 Suppose 0 i < n an p > For K Koo, n the following statements are equivalent: i The boy K is centere, ii The measure S p,i K, is even iii W p,i K, Q = W p,i K, Q, for all Q Koo n iv W p,i K, Q = W p,i K, Q, for Q = K Proof The results follow immeiately from Theorem 85 with φt = t p an p > This was prove by Lutwak [2] That iii implies that K is centrally symmetric, for the case p = an i = 0, was shown using other methos by Gooey [0] Acknowlegements This research was supporte by the National Natural Sciences Founation of China 37334

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