On p-radial Blaschke and harmonic Blaschke additions

In the paper, we first improve the radial Blaschke and harmonic Blaschke additions and introduce the p-radial Blaschke and p-harmonic Blaschke additions. Following this, Dresher type inequalities for the radial Blaschke-Minkowski homomorphisms with respect to p-radial Blaschke and p-harmonic Blaschke additions are established.


Notation and preliminaries
The setting for this paper is an 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 ρ(K, ·) : S n-1 → R defined for u ∈ S n-1 by ρ(K, u) = max{λ ≥ 0 : λu ∈ K}.
If ρ(K, ·) is positive and continuous, K will be called a star body. Let S n denote the set of star bodies in R n . Letδ denote the radial Hausdorff metric, i.e., if K, L ∈ S n , theñ δ(K, L) = |ρ(K, u)-ρ(L, u)| ∞ , where |·| ∞ denotes the sup-norm on the space of continuous functions C(S n-1 ).

Mixed intersection bodies
For K ∈ S n , there is a unique star body IK whose radial function satisfies, for u ∈ S n-1 , where v is (n -1)-dimensional dual volume. It is called the intersection body of K . The volume of the intersection body of K is given by (see [1]) The mixed intersection body of K 1 , . . . , K n-1 ∈ S n , denoted by I(K 1 , . . . , K n-1 ), is defined by whereṽ is the (n -1)-dimensional dual mixed volume. If K 1 = · · · = K n-i-1 = K, K n-i = · · · = K n-1 = L, then I(K 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 .

Improvement of the radial Blaschke addition
Let us recall the concept of radial Blaschke addition defined by Lutwak [1]. Suppose that K and L are star bodies in R n , the radial Blaschke addition denoted by K +L is a star body whose radial function is The dual Knesser-Süss inequality for the radial Blaschke addition was established by Lutwak [1]. If K, L ∈ S n , then with equality if and only if K and L are dilates.
In the section, we give a generalized concept of the radial Blaschke addition.

Definition 2.1
If K, L ∈ S n , 0 ≤ p < n-1 and λ, μ > 0 (not both zero), the p-radial Blaschke linear combination of K and L denoted by λ K + p μ L is a star body whose radial function is defined by , it is easy to see that When λ = μ = 1, the p-radial Blaschke combination becomes the p-radial Blaschke addition K + p L and Obviously, when p = 0, (2.4) becomes (2.1).
In the following, we define the dual mixed quermassintegral with respect to the p-radial Blaschke addition. First, we show two propositions. The following proposition follows immediately from (2.3) with L'Hôpital's rule.
The following proposition follows immediately from Proposition 2.2 and (1.6).

Definition 2.4
For 0 ≤ p < n -1, 0 ≤ i < n and K, L ∈ S n , the p-dual mixed quermassintegral of star bodies K and L, denoted by W p,i (K, L), is defined by Obviously, when K = L, W p,i (K, L) becomes the dual quermassintegral of star body K , i.e., , combining Hölder's integral inequality (see [3]) gives the following.
with equality if and only if K and L are dilates.
Taking i = 0 in (2.9), we have: If K, L ∈ S n and 0 ≤ p < n -1, then with equality if and only if K and L are dilates. In the following, we establish the Brunn-Minkowski inequality for the p-radial Blaschke addition.

Proposition 2.6
If K, L ∈ S n , 0 ≤ i < n and 0 ≤ p < n -1, then with equality if and only if K and L are dilates.
Proof From (2.3) and (2.7), it is easily seen that the p-dual mixed quermassintegral W p,i (K, L) is linear with respect to the p-radial Blaschke addition and together with inequality (2.9) shows that with equality if and only if K and L are dilates of Q. Take K + p L for Q in (2.12), recall that W p,i (Q, Q) = W i (Q), inequality (2.11) follows easy. Taking i = 0 in (2.11), we obtain that if K, L ∈ S n and 0 ≤ p < n -1, then with equality if and only if K and L are dilates. Taking p = 0 and i = 0 in (2.11), (2.11) becomes the well-known dual Knesser-Süss inequality (2.2).

Improvement of the harmonic Blaschke addition
Let us recall the concept of harmonic Blaschke addition defined by Lutwak [4]. Suppose that K and L are star bodies in R n , the harmonic Blaschke addition denoted by K+L is defined by Lutwak's Brunn-Minkowski inequality for the harmonic Blaschke addition was established (see [4]). If K, L ∈ S n , then with equality if and only if K and L are dilates.
In the section, we give an improved concept of the harmonic Blaschke addition.
Definition 3.1 For 0 ≤ i < n, p < i -1 and K, L ∈ S n , we define the p-harmonic Blaschke addition of star bodies K and L denoted by K+ p L and defined by The Brunn-Minkowski inequality for the p-harmonic Blaschke addition follows immediately from (1.6), (3.3) and Minkowski's integral inequality (see [3]). Proposition 3.2 If K, L ∈ S n , 0 ≤ i < n and p < i -1, then

4)
with equality if and only if K and L are dilates. (c) For all K, L ∈ S n and every ϑ ∈ SO(n),

Radial Blaschke-Minkowski homomorphisms
where SO(n) is the group of rotations in n dimensions.
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., [21]) (4. 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 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 [21]). In 2013, an L p Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms was established in [22]: If K and L are star bodies in R n and 0 < p < n -1, then with equality if and only if K and L are dilates. Taking p = 1, (4.3) reduces to (4.1).

Lemma 4.3 (see [5]) A map : S n → S n is a radial Blaschke-Minkowski homomorphism if and only if there is a measure
where M + (S n-1 ,ê) denotes the set of nonnegative zonal measures on S n-1 .
Obviously, a special case is the following: where i are integers. We now extend the integers i to real numbers, define the Blaschke-Minkowski homomorphism of order p of K .

Definition 4.4
Let K ∈ S n , the Blaschke-Minkowski homomorphism of order p of K , denoted by p K , is defined for all p ∈ R by ρ( p K, ·) = ρ(K, ·) n-1-p * μ. (4.6) This extended definition will be required to prove our main results.

Inequalities for the radial Blaschke-Minkowski homomorphism
Theorem 5.1 Let K, L ∈ S n . If 0 ≤ p < n -1 and i ≤ n -1 ≤ j ≤ n, then with equality if and only if p K and p L are dilates.

Remark 5.2
Taking j = n in (5.1) and noting that W n (K) = S n-1 dS(u) = nω n , (5.1) becomes the following inequality: If K, L ∈ S n , 0 ≤ p < n -1 and i ≤ n -1, then with equality if and only if p K and p L are dilates. Taking p = 0 in (5.1), (5.1) becomes the following inequality: If K, L ∈ S n and i ≤ n -1 ≤ j ≤ n, then with equality if and only if K and L are dilates.

Theorem 5.3
Let K, L ∈ S n . If 0 ≤ i < n, p < i -1 and k, j ∈ R satisfy j ≤ n -1 ≤ k ≤ n, then , (5.4) with equality if and only if p K and p L are dilates.

Remark 5.4
Taking k = n in (5.4) and noting that W n (K) = S n-1 dS(u) = nω n , (5.4) becomes the following inequality: If K, L ∈ S n , 0 ≤ i < n, p < i -1 and j ≤ n -1, then with equality if and only if p K and p L are dilates. Taking i = 0, j = 0 and k = n in (5.4), we have: If K, L ∈ S n and p < -1, then with equality if and only if p K and p L are dilates.

Dresher's inequalities for p-radial Blaschke and harmonic Blaschke additions
An extension of Beckenbach's inequality (see [3], p. 27) was obtained by Dresher [25] by means of moment-space techniques. We are now in a position to prove Theorem 5.1. The following statement is just a slight reformulation of it. Theorem 6.2 Let K, L ∈ S n . If 0 ≤ p < n -1 and s, t ∈ R satisfy s ≥ 1 ≥ t ≥ 0, then , (6.2) with equality if and only if p K and p L are dilates.
Equality holds if and only if the functions ρ( p K, u) and ρ( p L, u) are proportional. Taking s = ni and t = nj in Theorem 6.2, Theorem 6.2 becomes Theorem 5.1 stated in Section 5. If : S n × · · · × S n n-1 → S n is the mixed intersection operator I : S n × · · · × S n n-1 → S n in (6.2) and ns = i and nt = j, we obtain the following result: If K, L ∈ S n , 0 ≤ p < n -1 and i ≤ n -1 ≤ j ≤ n, then with equality if and only if I p K and I p L are dilates. Taking j = n in (6.5) and noting that W n (K) = S n-1 dS(u) = nω n , (6.5) becomes the following inequality: If K, L ∈ S n , 0 ≤ p < n -1 and i ≤ n -1, then with equality if and only if I p K and I p L are dilates.
We are now in a position to prove Theorem 5.3. The following statement is just a slight reformulation of it. Theorem 6.3 Let K, L ∈ S n . If 0 ≤ i < n, p < i -1 and s, t ∈ R satisfy s ≥ 1 ≥ t ≥ 0, then , (6.6) with equality if and only if p K and p L are dilates.
Proof From (3.3), we obtain Hence, from (4.6), we obtain By (1.6), we have and From (6.7), (6.8) and Lemma 6.1, we obtain   Taking s = nj and t = nk in Theorem 6.3, Theorem 6.3 becomes Theorem 5.3 stated in Section 5. If : S n × · · · × S n n-1 → S n is the mixed intersection operator I : S n × · · · × S n n-1 → S n in (6.6) and j = ns and k = nt, we obtain the following result: If K, L ∈ S n , 0 ≤ i < n, p < i -1 and j ≤ n -1 ≤ k ≤ n, then 1 W i (K+ p L) W j (I p (K+ p L)) W k (I p (K+ p L)) 1/(k-j) , (6.9) with equality if and only if I p K and I p L are dilates. Taking k = n in (6.9) and noting that W n (K) = S n-1 dS(u) = nω n , (6.9) becomes the following inequality: If K, L ∈ S n , 0 ≤ i < n, p < i -1 and j ≤ n -1, then W j (I p (K+ p L)) 1/(n-j) W i (K+ p L) ≤ W j (I p K) 1/(n-j) W i (K) + W j (I p L) 1/(n-j) W i (L) , (6.10) with equality if and only if I p K and I p L are dilates.

Conclusions
In the present study, we first revised and improved the concepts of radial Blaschke addition and harmonic Blaschke addition in an L p space. Following this, we established Dresher's inequalities (Brunn-Minkowski type) for the radial Blaschke-Minkowski homomorphisms with respect to the p-radial addition and the p-harmonic Blaschke addition.

Funding
The author's research is supported by the Natural Science Foundation of China (11371334).