Abstract
In this paper, we study the general dual Orlicz geominimal surface area by the general dual Orlicz mixed volume which was introduced by Gardner et al. (2019). We find the conditions to the existence of the general dual Orlicz-Petty body and hence prove the continuity of the general geominimal surface area in the Orlicz setting (2010 Mathematics Subject Classification: 52A20, 53A15).
1. Introduction
In the Euclidean space , we call a compact and convex subset a convex body if K has nonempty interior. Let be the family of all convex bodies in and be the family of the convex bodies containing the origin in their interiors. We use the standard notations and to denote the inner product and the Euclidean norm in . Let denote the volume of a set K and denote the unit ball in , i.e., ; the volume of is denoted by . Let be the unit sphere and σ denote the spheral measure.
The geominimal surface area and its extension play important roles in connecting the relative differential geometry with Minkowski geometry. The geominimal surface area was firstly introduced by Petty in 1974 (see [1]). The geominimal surface area was investigated by Lutwak (see [2]) and others (see, e.g., [3–5]). Recently, Lutwak et al. (see [6, 7]) developed the Orlicz–Brunn–Minkowski theory, a new extension of Brunn–Minkowski theory. Simultaneously, the geominimal surface area in Orlicz setting was studied in [8, 9]. The dual Brunn–Minkowski theory was introduced in [10, 11]. In 2011, Wang and Qi [12] introduced the geominimal surface area in dual theory which is dual to the geominimal surface area. Moreover, they established some affine isoperimetric inequalities, for example, the Blaschke Santaló-type inequality for the dual geominimal surface area. The dual Brunn–Minkowski theory in Orlicz setting has already been introduced by Gardner et al. [13, 14], Ye [15], and Zhu et al. [16]. For more results on the geominimal surface area, one can refer to [3, 4, 12, 17–20] and so on.
The Orlicz–Brunn–Minkowski theory for the general volume was established by Gardner et al. [21], and the general dual mixed volume in Orlicz setting was also introduced (see [21], p13), i.e., for two star bodies K and L:where is a continuous function, is also a continuous function, and denotes the radial function of K (see Section 2 for undefined and unexplained notations). There is a natural problem: whether there is a convex body with , such that it is a solution to the following problem:
Here, is the polar of M (see Section 2).
If for and , we show this problem is solvable. For instance, we obtain one of the main results.
Theorem 1. Let be a star body. If the function is increasing and , there is a convex body with , such that
In addition, if φ is convex, the solution T is unique.
Here, can be formulated by (1) and for , i.e.,
2. Background and Notation
For any set , it can be determined by its support function for . Obviously, if and if . A set K will be called a star set about the origin 0 if every line passes through 0 and meets K with only a line segment. Moreover, if K is a star set about 0, its radial function can be defined by the formula for all . If , one obtains
If the radial function of a star set K about the origin 0 is continuous and positive, then K is called a star body about the origin 0. Denote by the family of star bodies in endowed with radial metric; that is, the distance between is defined as , where is max-norm in Hausdorff metric space. Suppose there exist a sequence and a set such that , then the radial function is uniformly convergent to .
For a convex body , the polar of K is also a convex body in , that is,
It has been proved that if . Moreover, if , the following formulas hold:
We use the notation to denote the set of all invertible and linear transforms on and to denote the determinant of . Especially, . By and , we mean the transpose of and the inverse of the transpose of , respectively. From definitions of the radial function and the support function, we observe immediately that for , , and , one has
By formulas (7) and (8), if , we have
For any and , the harmonic radial combination is defined by (see [22])where defines the radial multiplication, i.e., . Obviously, the harmonic radial combination of two convex bodies is still a convex body according to formulas (7) and (10). For , the qth dual mixed volume of two star bodies is given by (see [2])
Letting , then . By the Hölder inequality, one has for (see [10])with equality iff K is a ball. If , the reverse inequality of (12) holds. Obviously,where is the volume of star body .
In convex geometry, the Blaschke selection theorem is very useful (see, e.g., [23, 24]), which means that if a sequence of convex bodies is bounded, there is a subsequence of and a convex set K such that converges to K.
Lemma 1 (see [2]). Suppose is a convergent sequence such that in the Hausdorff distance. If is bounded, then .
Lemma 2 (see [25]). If is a bounded sequence and the sequence is also bounded, there is a subsequence and a body such that . In addition, if , then .
3. Properties for the General Dual Mixed Volume
In this section, we will discuss some properties of the general dual Orlicz mixed volume, which is defined as follows [21]: Let ; if is continuous and , the general dual Orlicz mixed volume is
Here, denotes the family of all the continuous functions on interval , and is the set of such that . Let
In this paper, we will consider the function and for , that is
We will now prove the continuity of the general dual Orlicz mixed volume as follows.
Proposition 1. Let and be two sequences of star bodies such that and as . If and , then
Proof. Since and as , then and uniformly on , and thus, there are two constants such thatThen, for any and , we haveFrom the continuity of φ on interval , we haveHence, one hasThe next proposition is necessary to prove our main results.
Proposition 2. Let be a sequence and such that as . If is a sequence such that is bounded for and , then is bounded.
Proof. The boundedness of shows that there is a such that for all . We now assume is unbounded, that is, , where . Hence, there is a constant such that . Let and . Combining with continuity of the radial function, , and Fatou’s lemma, one hasLet ; then, one has a contradiction . Thus, is bounded.
4. Continuity of the General Dual Orlicz-Petty Body
In this section, we firstly define the general dual Orlicz geominimal surface area, denoted by , and prove the existence and uniqueness of the general dual Orlicz-Petty body. Moreover, we will also provide some properties of the general dual Orlicz-Petty body. Finally, the continuity of will be established.
Definition 1. Suppose is a star body. For and , define the general dual Orlicz geominimal surface area of K byThe affine invariance of the general dual Orlicz geominimal surface area can be shown in the following consequence.
Proposition 3. Let and . If and , then
Proof. For any , let . From Definition 1, together with formulas (8), (9), and (16), we haveWe now prove the existence of the general dual Orlicz-Petty body as follows.
Theorem 2. Suppose is a star body. If and , then there is a convex body such thatIn addition, if φ is convex, the body T is unique.
Proof. From Definition 1, there is a sequence with , such that for each and . We now prove that the sequence is bounded. Similar to Proposition 2, assume that is unbounded, let , and thus ; there exists a constant sequence and a constant D such that and as . Let and . By formula (16) and Fatou’s lemma, we haveThis will not hold if as . Hence, is bounded; then by the Blaschke selection theorem, there is one subsequence and one compact convex set T so that as ; it also implies that . Since , by Lemma 2, we have and by Lemma 1 we have . This together with Proposition 1 shows thatIf φ is convex, assume that there are two convex bodies such that and . Define a body T byObviouslyThus, T is a convex body and by formulas (7) and (10). By the Brunn–Minkowski inequality, we have with equality iff . Since both and are convex functions, we haveWith equality iff . Thus, if , we havewhich is a contradiction with the definition of . This finishes the proof.
Remark 1. The body T in Theorem 2 is usually called as the general dual Orlicz-Petty body and is denoted bySince T is unique if the function φ is convex by Theorem 2, the mapping contains only one element.
In the next theorem, we prove the continuity of the general geominimal surface area .
Theorem 3. Let and such that as . If and , then .
Proof. Let and such that as . For any real number , from Definition 1 and Proposition 1, there is a convex body so that andSince is arbitrary,Let for each , that is, . This, together with formula (35), implies that the sequence is bounded and thus the sequence is bounded by Proposition 2. Suppose is a subsequence so thatSince is uniformly bounded, then by Lemma 1, there is a subsequence and a convex body such that . According to Definition 1, Proposition 1, and Theorem 2, one hasCombining (35) and (37), one obtainsSimilar to the statement of Remark 1, the mapping contains only one element if the function φ is convex, so it can define an operator. Moreover, we can prove this operator is continuous as follows.
Theorem 4. Let and such that as . If and are convex, then .
Proof. Let and by the uniqueness of and when is convex. By the subsequence of such that as and Theorem 3, then there is a subsequence such thatThis implies that is bounded, and hence, is bounded by Proposition 2. Thus, according to Lemma 2, there are a subsequence and a body so that and . By Proposition 1 and formula (18), we haveOn the other hand, since , T satisfies and The convexity of , by Theorem 2, implies ; that is, . Hence, .
Data Availability
All data generated or analyzed during this study are included in this article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This paper was supported in part by NSFC (no. 11971005).