Orlicz Mean Dual Affine Quermassintegrals

Our main aim is to generalize the mean dual affine quermassintegrals to the Orlicz space. Under the framework of dual OrliczBrunn-Minkowski theory, we introduce a new affine geometric quantity by calculating the first Orlicz variation of the mean dual affine quermassintegrals and call it the Orlicz mean dual affine quermassintegral. The fundamental notions and conclusions of the mean dual affine quermassintegrals and the Minkowski and Brunn-Minkowski inequalities for them are extended to an Orlicz setting. The related concepts and inequalities of dual Orlicz mixed volumes are also included in our conclusions. The new Orlicz isoperimetric inequalities in special case yield the Lp-dual Minkowski inequality and Brunn-Minkowski inequality for the mean dual affine quermassintegrals, which also imply the dual Orlicz-Minkowski inequality and dual Orlicz-Brunn-Minkowski inequality.


Introduction
The radial addition  +  of star sets (compact sets that are star-shaped at  and contain )  and  can be defined by  +  = { +  :  ∈ ,  ∈ } , where  +  =  +  if , , and  are collinear and  +  = , otherwise, or by  ( + , ⋅) =  (, ⋅) +  (, ⋅) , where (, ⋅) denotes the radial function of star set , which is defined by  (, ) = max { ≥ 0 :  ∈ } , for  ∈  −1 , where  −1 is the surface of the unit sphere.Hints as to the origins of the radial addition can be found in [1, p. 235].If (, ⋅) is positive and continuous,  will be called a star body.Let S  denote the set of star bodies about the origin in R  .When combined with volume, radial addition gives rise to another substantial appendage to the classical theory, called the dual Brunn-Minkowski theory.Radial addition is the basis for the dual Brunn-Minkowski theory (see, e.g., [2][3][4][5][6][7][8][9][10] for recent important contributions).The original theory is originated from Lutwak [11].He introduced the concept of dual mixed volume which laid the foundation of the dual Brunn-Minkowski theory.The dual theory can count among its successes the solutions of the Busemann-Petty problem in [3,4,9,12,13].For  ̸ = 0,  ∈ R  , and ,  ∈ S  , the -radial addition  +  is defined by (see [14])
In recent years, a new extension of   -Brunn-Minkowski theory is to Orlicz-Brunn-Minkowski theory, initiated by Lutwak et al. [16,17].Gardner et al. [18] introduced the Orlicz addition for the first time, constructed a general framework for the Orlicz-Brunn-Minkowski theory, and made the relation to Orlicz spaces and norms clear.The Orlicz addition of convex bodies was also introduced from different angles and the   -Brunn-Minkowski inequality was extended to the Orlicz-Brunn-Minkowski inequality (see [19]).The Orlicz centroid inequality for star bodies was introduced in [20].The other articles advancing the theory can be found in literatures [7,[21][22][23][24][25].
The dual affine quermassintegrals were defined, for a convex body  ∈ S  , by letting Φ0 () fl (), Φ () fl   , and for 0 <  <  (see, e.g., [30], p. 515) where  , denotes the Grassmann manifold of dimensional subspaces in R  ,   denotes the gauge Haar measure on  , , vol  ( ∩ ) denotes the -dimensional volume of intersection of  on -dimensional subspace  ⊂ R  , and   denotes the volume of -dimensional unit ball.Gardner [31] showed the Brunn-Minkowski inequality for the dual affine quermassintegrals.If ,  ∈ S  and 0 ≤  ≤  − 1, then with equality if and only if  is a dilate of , modulo a set of measure zero.In analogy to (9), one may also define mean dual affine quermassintegrals by (see, e.g., [30], p. 516) for a convex body and 0 <  <  and by letting Φ 0 () fl () and Φ  () fl   .Here,  , denotes the space of the dimensional affine subspace in R  and ]  denotes the gauge Haar measure on  , .They are related to the dual affine quermassintegrals by (see [32], p. 373).
In the paper, our main aim is to generalize the mean dual affine quermassintegrals to the Orlicz space.Under the framework of dual Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity such as Orlicz mean dual affine quermassintegrals.The fundamental notions and conclusions of the mean dual affine quermassintegrals and the Minkowski and Brunn-Minkowski inequalities for the mean dual affine quermassintegrals are extended to an Orlicz setting.The new Orlicz-Minkowski and Brunn-Minkowski inequalities for the Orlicz mean dual affine quermassintegrals in special case yield the   -dual Minkowski inequality and Brunn-Minkowski inequalities for the mean dual affine quermassintegrals, which also imply the dual Orlicz-Minkowski inequality and Brunn-Minkowski inequalities for general volumes.
Obviously, the Orlicz mean dual affine quermassintegrals are an extension of the mean dual affine quermassintegrals; a very natural question is raised: is there a Minkowski type isoperimetric inequality for the Orlicz mean dual affine quermassintegrals?In Section 4, we give a positive answer to this question and establish the dual Orlicz-Minkowski inequality for the new affine geometric quantity.For  ∈ C, 0 <  ≤ , and ,  ∈ S  , we prove the Orlicz-Minkowski inequality for the Orlicz mean dual affine quermassintegrals.

(
Φ ,− (, ) If  is strictly convex, equality holds if and only if  and  are dilates.For  = , (17) becomes the following dual Orlicz-Minkowski inequality established by Zhu et al. [29]: If  is strictly convex, equality holds if and only if  and  are dilates.

Preliminaries
The setting for this paper is -dimensional Euclidean space R  .A body in R  is a compact set equal to the closure of its interior.For a compact set  ⊂ R  , we write () for the (dimensional) Lebesgue measure of  and call this the volume of .Associated with a compact subset  of R  , which is star-shaped with respect to the origin and contains the origin, its radial function is (, ⋅) :  −1 → [0, ∞), defined by (, ) = max{ ≥ 0 :  ∈ }.Note that the class (star sets) is closed under unions, intersection, and intersection with subspace.The radial function is homogeneous of degree −1; that is, (, ) =  −1 (, ), for all  ∈ R  and  > 0. Let δ denote the radial Hausdorff metric, as follows; if ,  ∈ S  , then (see, e.g., [30]) From the definition of the radial function, it follows immediately that for  ∈ () the radial function of the image  = { :  ∈ } of  is given by for all  ∈ R  .
The following result follows immediately from the definition of   -radial addition, with  ̸ = 0.
If ,  ∈ S  and  < 0, then with equality if and only if  and  are dilates.
Let ,  ∈ S  and  ≥ 1; the   -harmonic mixed volumes of star bodies  and  denotes Ṽ− (, ), defined by (see [35]) This integral representation (34), together with the Hölder inequality, yields Lutwak's   -dual Minkowski inequality as follows: If ,  ∈ S  and  ≥ 1, then with equality if and only if  and  are dilates.This integral representation (34), together with the definition of -harmonic addition, yields Lutwak's   -Brunn-Minkowski inequality for harmonic -addition (see [35]).If ,  ∈ S  and  ≥ 1, then with equality if and only if  and  are dilates.
The Orlicz harmonic linear combination on the case  = 2 is defined.

Orlicz Mean Dual Affine Quermassintegrals
In order to define Orlicz mean dual affine quermassintegrals, we need the following lemmas.
Specifically, we agreed on the following: In order to define the Orlicz mean dual affine quermassintegrals, we need also to calculate the first Orlicz variation of the mean dual affine quermassintegrals.
Proof.From Lemmas 7 and 9, we have, for  ∈ (), = Φ ,− (, ) . (60) Here, we point out the connections between the Orlicz mean dual affine quermassintegrals and the dual affine quermassintegrals.From (13) and in view of the connections between the mean dual affine quermassintegrals and the dual affine quermassintegrals, we have the following: for  ∈ C,  > 0, 0 <  ≤ , and ,  ∈ S  , We also need the following lemma to prove our main results.
Lemma 11 (Jensen's inequality).Let  be a probability measure on a space  and  :  →  ⊂ R is a -integrable function, where  is a possibly infinite interval.If  :  → R is a convex function, then If  is strictly convex, equality holds if and only if () is constant for -almost all  ∈  (see [38, p.165]).
On the other hand, suppose the equality holds in (63); then these three inequalities in the above proof must satisfy the equal sign.Since the first inequality in the above proof is the dual Orlicz-Minkowski inequality, Form the equality condition of dual Orlicz-Minkowski inequality, if the equality holds, then  ∩  and  ∩  must be dilates.The second inequality in the above proof is Jensen inequality.Proof.This follows immediately from (63) with  = .
The following uniqueness is a direct consequence of the Orlicz-Minkowski inequality for the Orlicz mean dual affine quermassintegrals.Proof.This follows immediately from Theorem 15 with  = .
Proof.This follows immediately from (90) with  = 1 and  = .) . ( If  is strictly convex, equality holds if and only if  and  are dilates. Proof.

⇐:
Let From Lemmas 4 and 7 and using the Orlicz-Brunn-Minkowski inequality (90), we obtain  This proof is complete.
From the proof of Theorem 19, we may see that Orlicz-Minkowski inequality for Orlicz mean dual affine quermassintegrals implies also Orlicz-Brunn-Minkowski inequality for the mean dual affine quermassintegrals.
Therefore ( +  ⋅ ) ∩  and ( ∩ ) +  ⋅ ( ∩ ) are the same star body in .Definition 6.If  ∈ C, 0 <  ≤ , and ,  ∈ S  , then Orlicz mean dual affine quermassintegral of  and , denoted by Φ ,− (, ), is defined by From the equality condition of Jensen inequality, if  is strictly convex and the equality holds, then vol  ( ∩ )/vol  ( ∩ ) must be a constant; this yields that  ∩  and  ∩  must be dilates.In this proof, the third inequality is obtained by applying the Hölder inequality.From the equality condition of Hölder inequality, this yields that equality holds and vol  ( ∩ ) and vol  ( ∩ ) must be proportional; namely,  ∩  and  ∩  are dilates.From the combinations of these equal conditions, it follows that equality in (63) holds, if  is strictly convex, and equality holds if and only if  and  are dilates.
and only if  and  are dilates.On the other hand, if taking  for , we similarly get Φ − () ≥ Φ − (), with equality if and only if  and  are dilates.Hence Φ − () = Φ − (), and  and  are dilates; it follows that  and  must be equal. and  are dilates.On the other hand, if we take  for , we similarly get Φ − () ≥ Φ − (), with equality if and only if  and  are dilates.Hence Φ − () = Φ − (), and  and  are dilates; it follows that  and  must be equal.