Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms

Abstract In 2006, Schuster introduced the radial Blaschke-Minkowski homomorphisms. In this article, associating with the star duality of star bodies and dual quermassintegrals, we establish Brunn-Minkowski inequalities and monotonic inequality for the radial Blaschke-Minkowski homomorphisms. In addition, we consider its Shephard-type problems and give a positive form and a negative answer, respectively.


Introduction and main results
is independent of ∈ − u S n 1 , where − S n 1 denotes the unit sphere in n .
The intersection body was introduced by Lutwak [2]. For each ∈ ≥ K n , 2 o n , the intersection body, IK, of K is an origin-symmetric star body whose radial function is defined by for all ∈ − u S n 1 . Here, ⊥ u is the ( − ) n 1 -dimensional hyperplane orthogonal to u and − V n 1 denotes the ( − ) n 1 -dimensional volume.
In 2006, based on the properties of intersection bodies, Schuster [3] introduced the radial Blaschke-Minkowski homomorphism, which is more general than intersection body operator as follows.
In 1999, Moszyńska [19] introduced the star duality of star bodies. For ∈ K o n , the star duality, ∘ K , of K is given by for every ∈ − u S n 1 . With the emergence of this notion, a number of characterizations and inequalities were established about star bodies, see [20][21][22][23][24][25][26].
We use ∘ K Ψ j to denote the star duality of the jth radial Blaschke-Minkowski homomorphism K Ψ j . The purpose of this article is to establish Brunn-Minkowski inequalities, monotonic inequality and consider Shephard-type problems for star duality of radial Blaschke-Minkowski homomorphisms. First, associated with the L p radial combination, we give the following Brunn-Minkowski inequality.
for − + < < j n p 1 0,   with equality if and only if K and L are homothetic. Here, + K L denotes the Minkowski sum of K and L, and * Q Φ j denotes the polar duality of Q Φ j . Comparing (1.5) with (1.6), we can feel that inequality (1.5) is an analogue of inequality (1.6). Note that the L p harmonic radial sum is just the − L p radial Minkowski sum of star bodies (see (2.1) and (2.2)). From this, if ≤ − p 1, we replace p by −p in Theorem 1.1, then inequality (1.4) yields the following Brunn-Minkowski inequality for the L p harmonic radial sum.
with equality if and only if K and L are dilates. Here, + ∼ − K L p denotes the L p harmonic radial addition of K and L.
Next, according to the L p harmonic Blaschke sum, another Brunn-Minkowski inequality for the star duality of radial Blaschke-Minkowski homomorphisms is established as follows.
o n , and reals i and j satisfy ≤ < − i j n 0 , 1, then for > p 0, equality holds if and only if K and L are dilates. Here, ∓ K L p denotes the L p Blaschke sum of K and L.
Furthermore, we consider the Shephard-type problems for the star duality of radial Blaschke-Minkowski homomorphisms and get a positive form and a negative answer, respectively.
equality holds if and only if = K L.
Theorem 1.4. Let Ψ i be an even ith radial Blaschke-Minkowski homomorphism, ∈ K o n and ≤ < − i n 0 1. If K is not origin-symmetric, then there exists ∈ L o n , such that In this article, the proofs of Theorems 1.1-1.4 are given in Section 3. In addition, in Section 3 we also give a monotonic inequality of star duality of the radial Blaschke-Minkowski homomorphisms.

Polar duality
If E is an arbitrary nonempty subset of n , the polar duality [1,27] , , 0 o n (both not zero) and real ≠ p 0, the L p radial Minkowski combination [27,28], n , of K and L is given by where "+ ∼ p " denotes the L p radial Minkowski addition, ∘ λ K denotes the L p radial scalar multiplication and , 0 (both not zero) and real ≥ p 1, then the L p harmonic radial combination [29], where "+ ∼ −p " denotes the L p harmonic radial addition. Obviously, − L p radial combination is the L p harmonic radial combination. Let , , 0 o n (both not zero) and real > p 0, the L p harmonic Blaschke combination [30,31], * ∓ * ∈ λ K μ L p o n , of K and L is given by where "∓ p " denotes the L p harmonic Blaschke addition, and * λ K denotes the L p harmonic Blaschke scalar multiplication and * = λ K λ K p 1 . When = p 1, (2.3) is the classical harmonic Blaschke combination.

Dual mixed quermassintegrals
Lutwak [32] gave the notion of dual mixed volume. For , the dual mixed volume, The Minkowski inequality [32] for the dual mixed quermassintegrals can be showed as follows.
with equality if and only if K is a dilate of L. If > − i n 1, inequality (2.8) is reverse. If = − i n 1, (2.8) becomes an equality.

General ith radial Blaschke bodies
,0 1 o n and ≥ λ μ , 0 (both not zero), the ith radial Blaschke combination [4], , of K and L is defined by (2.10) and = − L K in (2.9), writing and called ∇ K i τ  the general ith radial Blaschke body of K.
For the general ith radial Blaschke body [4], the following fact was also given.

Proofs of theorems
In this section, we will complete proofs of Theorems 1.1-1.4. In order to prove Theorem 1.1, we require the following lemmas.
, then by (2.6) and (2.8) we obtain , .   with ∈ (− ) τ 1, 1 , from the equality condition of the Minkowski integral inequality, we know that equality holds in (3.6) for ∈ (− ) τ 1, 1 if and only if K and −K are dilates, that is, K is origin-symmetric.
By the equality condition of (3.6), we know that with equality in   i.e., This is the desired inequality (3.7).
According to the equality condition of (2.8), with equality in (3.9) if and only if ∘ K Ψ j and ∘ L Ψ j are dilates, thus combined with equality conditions of (3.8) and (3.9), we know that equality holds in (3.7) if and only if = K L. □