Quotient for radial Blaschke-Minkowski homomorphisms by Chang-Jian Zhao (1), Wing-Sum Cheung (2)

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1 Bull. Math. Soc. Sci. Math. Roumanie Tome ) No Quotient for radial Blaschke-Minkowski homomorphisms by Chang-Jian Zhao 1) Wing-Sum Cheung 2) Abstract Some new inequalities for quotient function of quermassintegrals of the radial Blaschke- Minkowski homomorphisms are established. The results in special cases yield some of the recent results on inequalities of this type. Key Words: quotient function Radial Blaschke-Minkowski homomorphism L p - radial Minkowski addition L p -harmonic addition Mathematics Subject Classification: Primary 52A20. Secondary 46E30. 1 Introduction The setting for this paper is n-dimensional Euclidean space R n. We reserve the letter u for unit vectors and the letter B is reserved for the unit ball centered at the origin. The surface of B is S n 1. The volume of the unit n-ball is denoted by ω n. We use V K) for the n-dimensional volume of a body K. Associated with a compact subset K of R n which is star-shaped with respect to the origin is its radial function ρk ) : S n 1 R defined for u S n 1 by ρk u) = max{λ 0 : λu K}. If ρk ) is positive and continuous K will be called a star body. Let S n denote the set of star bodies in R n. Let δ denote the radial Hausdorff metric i.e. if K L S n then δk L) = ρ K ρ L where denotes the sup-norm on the space of continuous functions CS n 1 ). For K L S n and α β 0 Lutwak [14] defined the radial Blaschke linear combination α K β L as the star body whose radial function is given by ρα K β L ) n 1 = αρk ) n 1 βρl ) n 1. For K S n there is a unique star body IK whose radial function satisfies for u S n 1 ρik u) = vk E u ) where v is n 1)-dimensional volume and E u denotes the hyperplane orthogonal to u. It is called the intersection body of K. The volume of the intersection body of K is given by V IK) = 1 vk E u ) n dsu). n S n 1

2 148 Chang-Jian Zhao Wing-Sum Cheung The mixed intersection body of K 1... K n 1 S n IK 1... K n 1 ) is defined by ρik 1... K n 1 ) u) = ṽk 1 E u... K n 1 E u ) where ṽ is n 1)-dimensional dual mixed volume see below for the definition). If K 1 = = K n i 1 = K K n i = = K n 1 = L then IK 1... K n 1 ) is written as I i K L). If L = B then I i K L) is written as I i K and called the ith intersection body of K. For I 0 K we simply write IK. 1. Dual mixed volumes The radial Minkowski linear combination λ 1 K 1 λ r K r is defined by λ 1 K 1 λ r K r = {λ 1 x 1 λ r x r : x i K i i = 1... r} 1.1) for K 1... K r S n and λ 1... λ r R. It has the following important property see [14]) ρλk µl ) = λρk ) µρl ) 1.2) for K L S n and λ µ 0. For K 1... K r S n and λ 1... λ r 0 the volume of the radial Minkowski linear combination λ 1 K 1 λ r K r is a homogeneous polynomial of degree n in the λ i V λ 1 K 1 λ r K r ) = Ṽ K i1... K in )λ i1 λ in 1.3) where the sum is taken over all n-tuples i 1... i n ) whose entries are positive integers not exceeding r. If we require the coefficients of the polynomial in 1.3) to be symmetric in their arguments then they are uniquely determined. The coefficient Ṽ K i 1... K in ) is nonnegative and depends only on the bodies K i1... K in. It is called the dual mixed volume of K i1... K in. If K 1... K n S n then the dual mixed volume Ṽ K 1... K n ) can be represented in the form see [15]) Ṽ K 1... K n ) = 1 ρk 1 u) ρk n u)dsu). 1.4) n S n 1 If K 1 = = K n i = K K n i1 = = K n = L then the dual mixed volume is written as Ṽ i K L). If L = B then the dual mixed volume ṼiK L) = ṼiK B) is written as W i K). For K L S n the i-th dual mixed volume of K and L Ṽ i K L) can be extended to all i R by Ṽ i K L) = 1 ρk u) n i ρl u) i dsu). 1.5) n S n 1 Thus if K S n then for i R W i K) = 1 n S n 1 ρk u) n i dsu). 1.6)

3 Quotient for radial Blaschke-Minkowski homomorphisms Radial Blaschke-Minkowski homomorphisms and L p -radial addition Definition 2.1 [18]) A map Ψ : S n S n is called a radial Blaschke-Minkowski homomorphism if it satisfies the following conditions: a) Ψ is continuous. b) For all K L S n ΨK L) = ΨK) ΨL). c) For all K L S n and every ϑ SOn) ΨϑK) = ϑψk) where SOn) is the group of rotations in n dimensions. Radial Blaschke-Minkowski homomorphisms are important examples of star body valued valuations. Their natural duals Blaschke-Minkowski homomorphisms are an important notion in the theory of convex body valued valuations see e.g. [ ] and [ ]). In 2006 Schuster [18] established the following Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms of star bodies. If K and L are star bodies in R n then V ΨK L)) 1/nn 1) V ΨK) 1/nn 1) V ΨL) 1/nn 1) 2.1) with equality if and only if K and L are dilates. If K and L are star bodies in R n p 0 and λ µ 0 then λ K p µ L is the star body whose radial function is given by see e.g. [5]) ρλ K p µ L ) p = λρk ) p µρl ) p. 2.2) The addition p is called L p -radial addition. The L p dual Brunn-Minkowski inequality states: If K L S n and 0 < p n then V K p L) p/n V K) p/n V L) p/n with equality when p n if and only if K and L are dilates. The inequality is reversed when p > n or p < 0 see [5]). Very recently an L p Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms was established in [24]: If K and L are star bodies in R n and 0 < p < n 1 then V ΨK p L)) p/nn 1) V ΨK) p/nn 1) V ΨL) p/nn 1) 2.3) with equality if and only if K and L are dilates. Taking p = 1 2.3) reduces to 2.1). Theorem 2.2 see [18]) Let Ψ : S n S n be a radial Blaschke-Minkowski homomorphism. There is a continuous operator Ψ : S n S }{{ n S } n symmetric in its arguments such n 1 that for K 1... K m S n and λ 1... λ m 0 Ψλ 1 K 1 λ m K m ) = i 1...i n 1 λ i1 λ in 1 ΨK i1... K in 1 ). 2.4)

4 150 Chang-Jian Zhao Wing-Sum Cheung Clearly Theorem 2.2 generalizes the notion of radial Blaschke-Minkowski homomorphisms. We call Ψ : S n S n S n mixed radial Blaschke-Minkowski homomorphism induced by Ψ. Mixed radial Blaschke-Minkowski homomorphisms were first studied in more detail in [19-20]. If K 1 = = K n i 1 = K K n i = = K n 1 = L we write Ψ i K L) for ΨK... K L... L). If K }{{}}{{} 1 = = K n i 1 = K K n i = = K n 1 = B we write Ψ i K n i 1 i for ΨK... K B... B) and call Ψ i K the mixed Blaschke-Minkowski homomorphism of } {{ } n i 1 } {{ } i order i of K. Lemma 2.3 see [18]) A map Ψ : S n S n is a radial Blaschke-Minkowski homomorphism if and only if there is a measure µ M S n 1 ê) such that ρψk ) = ρk ) n 1 µ 2.5) where M S n 1 ê) denotes the set of nonnegative zonal measures on S n 1. For the mixed radial Blaschke-Minkowski homomorphism induced by Ψ Schuster [18] proved that ρψk 1... K n 1 ) ) = ρk 1 ) ρk n 1 ) µ. We now define the mixed Blaschke-Minkowski homomorphism of order i of K for all i R by ρψ i K ) = ρk ) n 1 i µ. 2.6) This extended definition will be required in the following. In 2013 the quotient function of the volumes was first introduced in [27]: Let K and D be star bodies in R n then the dual quermassintegral quotient function of star bodies K and D Q WijKD) defined by Q WijKD) = W i K) i j R. W j D) The aim of this paper is to establish the following inequalities for quermassintegral of quotient function of radial Blaschke-Minkowski homomorphisms with respect to L p -radial addition. Theorem 2.4 Let K L D D S n. If p 0 and i n 1 j n then ) 1/j i) ) 1/j i) ) 1/j i) Wi Ψ n 1 p K p L)) Wi Ψ n 1 p K) Wi Ψ n 1 p L) W j Ψ n 1 p D p D )) W j Ψ n 1 p D) W j Ψ n 1 p D ) 2.7) with equality if and only if Ψ n 1p K and Ψ n 1p L are dilates and Ψ n 1p D and Ψ n 1p D are dilates and Wi Ψ n 1 p K) 1/n i) W i Ψ n 1 p L) 1/n i)) = µ Wj Ψ n 1 p D) 1/n j) W j Ψ n 1 p D ) 1/n j)) for some constant.

5 Quotient for radial Blaschke-Minkowski homomorphisms 151 Remark 2.4 Putting D = K and D = L in 2.7) 2.7) becomes an inequality established in [28]. Let D = K and D = L and putting j = n in 2.7) 2.7) becomes the following inequality: If K L S n p 0 and i n 1 then W i Ψ n 1 p K p L)) 1/n i) W i Ψ n 1 p K) 1/n i) W i Ψ n 1 p L) 1/n i) with equality if and only if Ψ n 1 p K and Ψ n 1 p L are dilates. Taking p = n 1 in 2.7) 2.7) reduces to the following inequality: If K L D D S n and i n 1 j n then ) 1/j i) ) 1/j i) ) 1/j i) Wi ΨK n 1 L)) Wi ΨK) Wi ΨL) 2.8) W j ΨD n 1 D )) W j ΨD) W j ΨD ) with equality if and only if ΨK and ΨL are dilates and ΨD and ΨD are dilates and Wi ΨK) 1/n i) W i ΨL) 1/n i)) = µ Wj ΨD) 1/n j) W j ΨD ) 1/n j)) for some constant. Taking j = n in 2.8) 2.8) reduces to the following inequality: If K L S n and i n 1 then W i ΨK n 1 L)) 1/n i) W i ΨK) 1/n i) W i ΨL) 1/n i) with equality if and only if ΨK and ΨL are dilates. 3 Radial Blaschke-Minkowski homomorphisms and L p -harmonic addition If K L S n and λ µ 0 not both zero) then for p 1 the L p -harmonic combination λ K ˆ p µ L S n was defined by ρλ K ˆ p µ L ) p = λρk u) p µρl u) p. 3.1) In 1996 Lutwak [17] established an L p -Brunn-Minkowski inequality for harmonic addition. If K L S n and p 1 then V K ˆ p L) p/n V K) p/n V L) p/n 3.2) with equality if and only if K and L are dilates. Another aim of this paper is to establish the following Dresher type inequality for radial Blaschke-Minkowski homomorphisms with respect to L p -harmonic addition. Theorem 3.1 Let K L D D S n. If p 1 and i n 1 j n then ) 1/j i) ) 1/j i) ) 1/j i) Wi Ψ n 1p K ˆ p L)) Wi Ψ n 1p K) Wi Ψ n 1p L) W j Ψ n 1p D ˆ p D )) W j Ψ n 1p D) W j Ψ n 1p D ) 3.3) with equality if and only if Ψ n 1p K and Ψ n 1p L are dilates and Ψ n 1p D and Ψ n 1p D are dilates and Wi Ψ n 1p K) 1/n i) W i Ψ n 1p L) 1/n i)) = µ Wj Ψ n 1p D) 1/n j) W j Ψ n 1p D ) 1/n j))

6 152 Chang-Jian Zhao Wing-Sum Cheung for some constant. Remark 3.2 Putting D = K and D = L in 3.3) 3.3) becomes an inequality established in [28]. Let D = K and D = L and putting j = n in 3.3) 3.3) becomes the following inequality: If K L S n p 1 and i n 1 then W i Ψ n 1p K ˆ p L)) 1/n i) W i Ψ n 1p K) 1/n i) W i Ψ n 1p L) 1/n i) with equality if and only if Ψ n 1p K and Ψ n 1p L are dilates. Taking i = 0 in 3.3) 3.3) becomes the following inequality: If K L D D S n and n 1 j n then ) 1/j ) 1/j V Ψ n 1p K ˆ p L)) V Ψ n 1p K) W j Ψ n 1p D ˆ p D )) W j Ψ n 1p D) ) 1/j V Ψ n 1p L) 3.3) W j Ψ n 1p D ) with equality if and only if Ψ n 1p K and Ψ n 1p L are dilates and Ψ n 1p D and Ψ n 1p D are dilates and V Ψ n 1p K) 1/n i) V Ψ n 1p L) 1/n i)) = µ Wj Ψ n 1p D) 1/n j) W j Ψ n 1p D ) 1/n j)) for some constant. 4. Inequalities for radial Blaschke-Minkowski homomorphisms An extension of Beckenbach s inequality see [3] p.27) was obtained by Dresher [4] by means of moment-space techniques: If p 1 r 0 f g 0 and φ is a distribution function then f g) p ) 1/p r) dφ f p ) 1/p r) dφ g p ) 1/p r) dφ f g)r dφ f r. dφ gr dφ Recently a new Dresher type inequality was derived in [26] as follows. Lemma 4.1 Let E be a bounded measurable subset of R n let φ be a distribution function and let f 1 f 2 g 1 g 2 : E R. If p 1 r 0 then Ef 1 f 2 ) p dφ E g 1 g 2 ) r dφ ) 1 p r E f p 1 dφ E gr 1 dφ ) 1 p r E f p 2 dφ E gr 2 dφ ) 1 p r 4.1) with equality if and only if f 1 = k 1 f 2 g 1 = k 2 g 2 and f 1 p f 2 p ) = µ g 1 r g 2 r ) where k 1 k 2 µ are constants. We prove now Theorem 3.1. The following statement is just a slight reformulation of it: Theorem 4.2 Let K L D D S n. If p 1 and s t R satisfy s 1 t 0 then ) 1/s t) ) 1/s t) ) 1/s t) Wn s Ψ n 1p K ˆ p L)) Wn s Ψ W n t Ψ n 1p D p D n 1p K) Wn s Ψ n 1p L) )) W n t Ψ n 1p D) W n t Ψ n 1p D 4.2) ) with equality if and only if Ψ n 1p K and Ψ n 1p L are dilates and Ψ n 1p D and Ψ n 1p D are dilates and Wn s Ψ n 1p K) 1/s W n s Ψ n 1p L) 1/s) = µ Wn t Ψ n 1p D) 1/t W n t Ψ n 1p D ) 1/t).

7 Quotient for radial Blaschke-Minkowski homomorphisms 153 Proof From 3.1) we have for p 1 ρk ˆ p L ) p µ = ρk ) p µ ρl ) p µ where µ is the generating measure of Ψ from Lemma 2.3. Hence from 2.6) we obtain Therefore by 1.6) we have and ρψ n 1p K ˆ p L) ) = ρψ n 1p K ) ρψ n 1p L ). W n s Ψ n 1p K ˆ p L)) = 1 n W n t Ψ n 1p D ˆ p D )) = 1 n From 4.3) 4.4) and Lemma 4.1 we obtain S n 1 ρψ n 1p K u) ρψ n 1p L u)) s dsu) 4.3) S n 1 ρψ n 1p D u) ρψ n 1p D u)) t dsu). 4.4) ) 1/s t) Wn s Ψ n 1p K ˆ p L)) ρψ S = n 1 n 1p K u) ρψ n 1p L u)) s dsu) W n t Ψ n 1p D p D )) ρψ S n 1 n 1p D u) ρψ n 1p D u)) t dsu) ρψ n 1p K u) s ) 1/s t) dsu) ρψ Sn 1 n 1p L u) s dsu) Sn 1 ρψ S n 1 n 1p D u) t dsu) ρψ S n 1 n 1p D u) t dsu) ) 1/s t) ) 1/s t) Wn s Ψ n 1p K) Wn s Ψ n 1p L) =. W n t Ψ n 1p D) W n t Ψ n 1p D ) ) 1/s t) ) 1/s t) From the equality condition of Lemma 4.1 equality in 4.2) holds if and only if the functions ρψ n 1p K u) and ρψ n 1p L u) are proportional and ρψ n 1p D u) and ρψ n 1p D u) are proportional and Wn s Ψ n 1p K) 1/s W n s Ψ n 1p L) 1/s) = µ Wn t Ψ n 1p D) 1/t W n t Ψ n 1p D ) 1/t). Taking s = n i and t = n j in Theorem 4.2 Theorem 4.2 becomes Theorem 3.1 stated in Section 3. If Ψ : S n S }{{ n } n 1 S n is the mixed intersection operator I : S n S }{{ n S } n in n 1 4.2) and n s = i and n t = j we obtain the following result: If K L D D S n p 1 and i n 1 j n then ) 1/j i) ) 1/j i) ) 1/j i) Wi I n 1p K ˆ p L)) Wi I n 1p K) Wi I n 1p L) W j I n 1p D p D )) W j I n 1p D) W j I n 1p D ) 4.5)

8 154 Chang-Jian Zhao Wing-Sum Cheung with equality if and only if I n 1p K and I n 1p L are dilates I n 1p D and I n 1p D are dilates and Wi I n 1p K) 1/n i) W i I n 1p L) 1/n i)) = µ Wj I n 1p D) 1/n j) W j I n 1p D ) 1/n j)). Taking j = n in 4.5) and noting that W n K) = S n 1 dsu) = nω n 4.5) becomes the following inequality: If K L S n p 1 and i n 1 then W i I n 1p K ˆ p L)) 1/n i) W i I n 1p K) 1/n i) W i I n 1p L) 1/n i) with equality if and only if I n 1p K and I n 1p L are dilates. Theorem 4.3 Let K L D D S n. If p 0 and s t R satisfy s 1 t 0 then ) 1/s t) ) 1/s t) ) 1/s t) Wn s Ψ n 1 p K p L)) Wn s Ψ n 1 p K) Wn s Ψ n 1 p L) W n t Ψ n 1 p D p D )) W n t Ψ n 1 p D) W n t Ψ n 1 p D ) 4.6) with equality if and only if Ψ n 1p K and Ψ n 1p L are dilates and Ψ n 1p D and Ψ n 1p D are dilates and Wn s Ψ n 1p K) 1/s W n s Ψ n 1p L) 1/s) = µ Wn t Ψ n 1p D) 1/t W n t Ψ n 1p D ) 1/t). Proof From 2.2) we have for p 0 Hence from 2.6) we obtain By 1.6) we have and ρk p L ) p µ = ρk ) p µ ρl ) p µ. ρψ n 1 p K p L) ) = ρψ n 1 p K ) ρψ n 1 p L ). W n s Ψ n 1 p K p L)) = 1 n W n t Ψ n 1 p D ˆ p D )) = 1 n From 4.7) 4.8) and Lemma 4.1 we obtain S n 1 ρψ n 1 p K u) ρψ n 1 p L u)) s dsu) 4.7) S n 1 ρψ n 1 p D u) ρψ n 1 p D u)) t dsu). 4.8) ) 1/s t) Wn s Ψ n 1 p K p L)) ρψ S = n 1 n 1 p K u) ρψ n 1 p L u)) s dsu) W n t Ψ n 1 p D p D )) ρψ S n 1 n 1 p D u) ρψ n 1 p D u)) t dsu) ρψ n 1 p K u) s ) 1/s t) dsu) ρψ Sn 1 S n 1 n 1 p L u) s dsu) ρψ S n 1 n 1 p D u) t dsu) ρψ S n 1 n 1 p D L u) t dsu) ) 1/s t) ) 1/s t)

9 Quotient for radial Blaschke-Minkowski homomorphisms 155 ) 1/s t) ) 1/s t) Wn s Ψ n 1 p K) Wn s Ψ n 1 p L) =. W n t Ψ n 1 p D) W n t Ψ n 1 p D ) From the equality condition of Lemma 4.1 equality in 4.6) holds if and only if the functions ρψ n 1p K u) and ρψ n 1p L u) are proportional and ρψ n 1p D u) and ρψ n 1p D u) are proportional and Wn s Ψ n 1p K) 1/s W n s Ψ n 1p L) 1/s) = µ Wn t Ψ n 1p D) 1/t W n t Ψ n 1p D ) 1/t). Taking s = n i and t = n j in Theorem 4.3 Theorem 4.3 becomes Theorem 2.4 stated in Section 2. If Ψ : S n S }{{ n S } n is the mixed intersection operator I : S n S n S }{{} n in n 1 n 1 4.6) and i = n s and j = n t we obtain the following result: If K L D D S n p 0 and i n 1 j n then ) 1/j i) ) 1/j i) ) 1/j i) Wi I n 1 p K p L)) Wi I n 1 p K) Wi I n 1 p L) W j I n 1 p D p D )) W j I n 1 p D) W j I n 1 p D ) 4.9) with equality if and only if I n 1 p K and I n 1 p L are dilates and I n 1 p D and I n 1 p D are dilates and Wn s I n 1 p K) 1/s W n s I n 1 p L) 1/s) = µ Wn t I n 1 p D) 1/t W n t I n 1 p D ) 1/t). Taking j = n in 4.9) and noting that W n K) = S n 1 dsu) = nω n 4.9) becomes the following inequality: If K L S n p 0 and i n 1 then W i I n 1 p K p L)) 1/n i) W i I n 1 p K) 1/n i) W i I n 1 p L) 1/n i) 4.10) with equality if and only if I n 1 p K and I n 1 p L are dilates. Taking p = n 1 in 4.10) 4.10) reduces to the following inequality: If K L S n and i n 1 then W i IK n 1 L)) 1/n i) W i IK) 1/n i) W i IL) 1/n i) with equality if and only if IK and IL are dilates. Taking p = n 1 in 4.9) 4.9) reduces to the following inequality: If K L D D S n and i n 1 j n then ) 1/j i) ) 1/j i) ) 1/j i) Wi IK n 1 L)) Wi IK) Wi ΨL) W j ID n 1 D )) W j ID) W j ΨD ) with equality if and only if IK and IL are dilates and ID and ID are dilates and Wn s IK) 1/s W n s IL) 1/s) = µ Wn t ID) 1/t W n t ID ) 1/t). Acknowledgement. The first author expresses his grateful thanks to Prof. Gangsong Leng for his many suggestions. This research was supported by the National Natural Science Foundation of China ) and by HKU Seed Grant for Basic Research.

10 156 Chang-Jian Zhao Wing-Sum Cheung References [1] J. Abardia A. Bernig Projection bodies in complex vector spaces Adv. Math ). [2] S. Alesker A. Bernig F. E. Schuster Harmonic Analysis of translation invariant valuations Geom. Funct. Anal ). [3] E. F. Beckenbach R. Bellman Inequalities Springer-Verlag Berlin-Göttingen Heidelberg 1961). [4] M. Dresher Moment space and inequalities Duke. Math. J ). [5] R. J. Gardner The Brunn-Minkowski inequality Bull. Amer. Math. Soc ). [6] C. Haberl Star body valued valuations Indiana Univ. Math. J ). [7] C. Haberl F. E. Schuster General L p affine isoperimetric inequalities J. Diff. Geom ). [8] C. Haberl L p intersection bodies Adv. Math ). [9] C. Haberl M. Ludwig A characterization of L p intersection bodies Int. Math. Res. Not. Art ID pages. [10] M. Kiderlen Blaschke- and Minkowski-Endomorphisms of convex bodies Trans. Amer. Math. Soc ). [11] M. Ludwig Minkowski valuations Trans. Amer. Math. Soc ) ). [12] M. Ludwig General affine surface areas Adv. Math ). [13] M. Ludwig Minkowski areas and valuations J. Diff. Geom ). [14] E. Lutwak Intersection bodies and dual mixed volumes Adv. Math ). [15] E. Lutwak Dual mixed volumes Pacific J. Math ). [16] E. Lutwak Mixed projection inequalities Trans. Amer. Math. Soc. 2871) ). [17] E. Lutwak The Brunn-Minkowski-Firey theory II. Affine and geominimal surface areas. Adv. Math ). [18] F. E. Schuster Volume inequalities and additive maps of star bodies Mathematika ). [19] F. E. Schuster Valuations and Busemann-Petty type problems Adv. Math ).

11 Quotient for radial Blaschke-Minkowski homomorphisms 157 [20] F. E. Schuster Convolutions and multiplier transformations of star bodies Trans. Amer. Math. Soc ). [21] F. E. Schuster Crofton measures and Minkowski valuations Duke Math. J ). [22] F. E. Schuster T. Wannerer GLn) contravariant Minkowski valuations Trans. Amer. Math. Soc ). [23] T. Wannerer GLn) equivariant Minkowski valuations Indiana Univ. Math. J ). [24] W. Wang L p Brunn-Minkowski type inequalities for Blaschke-Minkowski homomorphisms Geom. Dedicata ). [25] C. J. Zhao On Blaschke-Minkowski homomorphisms Geom. Dedicata ). [26] C. J. Zhao Quotient functions of dual quermassintegrals Quaestions Math. 382) ). [27] C. J. Zhao On Blaschke-Minkowski homomorphisms Indagationes Math ). [28] C. J. Zhao W. S. Cheung Dresher type inequalities for radial Blaschke- Minkowski homomorphisms Moscow J. Math in press. Received: Accepted: ) Department of Mathematics China Jiliang University Hangzhou P. R. China chjzhao@163.com chjzhao@aliyun.com 2) Department of Mathematics The University of Hong Kong Pokfulam Road Hong Kong wscheung@hku.hk