Abstract
Motivated by Lutwak et al.’s -dual curvature measures, we introduce the concept of -curvature measures. This new -curvature measure is an extension of the classical surface area measure, -surface area measure, and curvature measure. In this paper, we first prove some properties of the -curvature measure. Next, using the -curvature measure, we define the -mixed volume which includes -mixed volume as the special cases. Further, the Minkowski-type inequality related -mixed volume and the uniqueness of the solution for the - Minkowski problem are obtained. Finally, we propose several problems that need to be studied further.
1. Introduction
Surface area measure and integral curvature measure are two important measures in classical Brunn-Minkowski theory. Minkowski problem describing surface area measure and Aleksandrov problem describing integral curvature are two famous problems. As a generalization, -surface area measure and -integral curvature are defined in [1, 2], respectively. At the same time, the hyperbolic measure as the curvature measure of dual Fiedler is constructed in [3]. Lutwak et al. introduce -dual curvature measure in [4], which is a generalization of the dual curvature, -surface area measure and -integral curvature. -dual mixed volume (also known as -dual mixed volume) is defined by [4] and Minkowski inequality is established. Furthermore, they study the -dual Minkowski problem of -dual curvature measure by reference to [5].
Inspired by Lutwak et al.’s -dual curvature measure, a new concept of -curvature measure is introduced in this paper. It includes classical surface area measure, -surface area measure and curvature measure. In this paper, we first prove some properties of -curvature measure. Next, based on -curvature measure, we define -mixed volume, which includes -mixed volume as a special case. Furthermore, the Minkowski inequality for -mixed volume and the uniqueness of the solution for -Minkowski problem are obtained. Finally, some problems which need further study are put forward.
Let represent the set of convex bodies in -dimensional Euclidean (compact convex subsets with nonempty embedding) space , for convex bodies containing the origin inside in , we write . Set said centered on the origin of the unit sphere, surface written as , in . represents the dimensional volume of the body and writes .
For , its support function, , is defined by (see [6]) where denotes the standard inner product of and .
For and (not both zero), the Minkowski combination, , of and is defined by the following:
i.e., .
The surface area measure of can be defined by the following: for any . From Equation (3), the Minkowski’s first mixed volume of and is given as follows:
The mixed volume generalizes the concepts of volume, surface area, and mean width.
We say that has a positive continuous curvature function , if for all , where is spherical Lebesgue measure. Clearly, Equations (4) and (5) imply the following:
Let . Using the -Minkowski conbinations (see Equation (60)), Lutwak [2] defined the -surface area measure of a convex body , namely, for each ,
For , the -mixed volume is given by the following (see [4]):
We say that has a positive continuous -curvature function , if the integral representation for all . For with a positive continuous curvature functions, it follows from Equation (8) and Equation (9) that
The -Minkowski inequality of the -mixed volume is (see [2, 7]) that for , with equality for if and only if and are dilates, for and if and only if and are homothetic.
According to Equation (10), the curvature function of is the Radon-Nikodym derivative of -surface area measure with respect to the spherical Lebesgue measure. The integral of -curvature function (raised to an appropriate power) over the unit sphere is the -affine surface area, which is an important research point of affine geometry and valuation theory, see, e.g., [8–24]. The -Minkowski problem (see [2]) is a necessary and sufficient condition to find a given measure such that it is only the -surface area measure of a convex body. Solving the -Minkowski problem requires solving a degenerate singular Monge-Ampère-type equation on the unit sphere. The -Minkowski problem has been solved for , see [2, 25, 26], but critical cases for remain open, see, e.g., [25, 27–31]. For its applications, see [5, 7, 27, 32–35].
A star body is a compact star-shaped set about the origin whose radial function is defined by the following: for . If is positive and continuous, will be called a star body. Denote the set of star bodies in by . Obviously, .
The dual Brunn-Minkowski theory is the theory of dual mixed volumes of star bodies. For , the -th dual mixed volume, , of is defined by the following: where the integration is with respect to spherical Lebesgue measure. For , the -th dual volume of is defined by . The -th dual volume is important in geometric tomography, one of the reasons that is that for integers and each , where denotes volume in , denote the Grassmann manifold of -dimensional subspaces of , the integration is with respect to the rotation invariant probability measure on and constant is trivially determined by taking to be .
For the real , the -th dual curvature of is a Borel measure on the unit sphere, which can be defined in [3] by using the variational formula: for every . Similar to the critical role as -surface area measures playing in the Brunn-Minkowski theory, dual curvature measures is a central concept within the dual Brunn-Minkowski theory.
The singularity case of dual volume leads to dual entropy of star body. For , the dual entropy can be defined as follows:
The -integral curvature, , of (see [1]) can be defined by a variational formula: for all , where is the polar body of is given by .
In [4], Lutwak et al. introduced -dual curvature measures, which are a generalization of dual curvatures, -surface area measure and -integral curvatures. For , and , the -dual curvature measure, , is the Borel measure on defined by the following: for each continuous , where is the radial Gauss map (see Section 2 for details).
-dual mixed volume (also known as -dual mixed volume) is defined by Lutwak et al. [4] using the -dual curvature:
For , , and , the -dual mixed volume is defined by the following:
By Equation (18), the -dual mixed volume has the following integral formula:
Specifically, , namely, and for ,
For the -dual mixed volumes, the related Minkowski inequality is given in [4]. Suppose , if and , then with equality when if and only if and are dilates; while when and , with equality if and only if and are dilates; while when and , with equality if and only if and are homothetic.
In [4], the authors studied the -dual Minkowski problems for -dual curvature measures. The results of -dual Minkowski problem caught many attentions, for example, see [3, 27, 36–42]. In addition, based on the -dual mixed volumes, Ma et al. studied -John ellipsoids in [43], which contain the classical John ellipsoid and the -John ellipsoids. They also solved two involving optimization problem about the -dual mixed volumes for all . A different extension of the -John ellipsoid was considered by Li et al. in [44].
In this paper, motivated by Lutwak et al.’s works in [4], we introduce the following -curvature measures which is a new curvature measure.
Definition 1. For and , we define the -curvature measure by the following: for each continuous .
According to Definition 1, the -curvature measure has the following integral expression.
Property 2. Suppose . If , then for each Borel set . Here, and is the supporting hyperplane to with outer normal vector .
Property 3. Suppose . If , then for each Borel set ,
Among them, represents the -dimensional Hausdorff measure, and represents the regular radial vector of , as well as represents the reverse spherical image of .
The -curvature measures unify the surface area measures, -surface area measures and curvature measures, as well as other measures. In particular, for and , the -surface area measures and the -th curvature measures (see Section 3 for its definition) are special cases of the -curvature measures:
According to the -curvature measures, we now define the notion of the -mixed volumes which unifies -mixed volumes and dual-mixed volumes.
Definition 4. For and , the -mixed volume, , of and (with respect to ) is defined by the following:
The following variational formula is an extension of Equations (3) and (7).
Theorem 5. If reals and , then the -mixed volume via the variational formula of and (with respect to ) by the following:
Using Equation (24), the -mixed volume can be written by the following integral formula:
It will be shown that the -mixed volume (Equation (8)) is the special case of the -mixed volumes of convex bodies, i.e.,
The Minkowski-type inequality for -mixed volume is as follows:
Theorem 6. Let and . Then, with equality if and only if are dilates when and are homothetic when .
For , we say that the convex body with respect to has a positive continuous -curvature function , if for all . From Equations (33) and (38), we get that for with a positive continuous curvature functions and a fixed ,
For and , the normalized power function can be defined by the following:
For and , the normalized -mixed volume is defined by the following:
Note that for , we have , while for the normalized -mixed volume is not just multiplied by a constant but it can be considered from the mixed entropy (see Section 2 for details).
Another aim of this paper is to show that for and , there exists a variational formula that defines the -curvature measure by the following: for every . This plays a key role to solve the associated Minkowski-type problems using a variational method.
Associated with -curvature measures, -Minkowski problem related to -curvature measure asks: For a given Borel measure on a sphere, what are the necessary and sufficient conditions for the existence of a convex body whose -curvature measure is ? The uniqueness of the problem is to ask to what extent is a convex body uniquely determined by its -curvature measure?
The new -Minkowski problem is equivalent to a degenerate singular Monge-Ampère equation on : For fixed , where is the given “data” function, is the unknown function, and is the identity map. Here, and denote the gradient vector and the Hessian matrix of , respectively, with respect to an orthonormal frame on , and is the identity matrix. If we assume that the range of the gradient function is , then is also an unknown function related to .
Finally, we propose some problems that need further study, i.e., -affine surface area problem, -geominimal surface area problem and -John ellipsoid problem.
2. Preliminaries
2.1. Basics in Convex Geometry
We work in the -dimensional Euclidean space . For , we use to denote the standard inner product of and , and to denote the Euclidean norm of . For , we will use both and to abbreviate .
We denote by the family of continuous functions defined on as endowed with the topology induced by the max-norm: , for .
For the support function, we know that for and ,
Generally, for , the image satisfies that for , where denotes the transpose of .
Since the support function is positive homogeneous of degree , we can restricted it on the unit sphere. For convex bodies , their Hausdorff metric is given by the following:
At the point where is differentiable, the gradient of in is as follows: where denotes the gradient of on with respect to the standard metric of .
For the radial function, we see that for , and ,
Using the radial function, the volume of can be expressed as follows:
For , the polar body of is defined by the following:
From this definition, we get that for , and for ,
For , the Minkowski function of is defined by the following:
Obviously, it is a continuous function on , and
In the whole process, will represent a closed set that cannot be contained in any of the closed hemispheres of . Wulff shape , a continuous function , also known as of the Aleksandrov body, is defined by the following:
If , then it is easily seen that
Assume that the function is continuous. Since is assumed to be closed, and is continuous, we have is a compact set in . The convex hull generated by , is compact as well (see Schneider [40], Theorem 1.1.11). Since is not contained in any closed hemisphere of , we get that contains the origin in its interior; namely, . Obviously, if ,
The following lemma will be required.
Lemma 7 (see [3]). Let be a closed set that is not contained in any closed hemisphere of . Let be continuous. Then, the Wulff shape determined by and the convex hull generated by the function are polar reciprocals of each other; namely, Let and . The -Minkowski combination is the convex body whose support function is given by the following (see [2]):
From Equation (53), we can extend the -Minkowski combinations to the cases of .
Let . For , and such that is a strictly positive function on , Lutwak et al. [4] defined the -Minkowski combination by the following:
When , define by the following:
Note that is defined for all , since are strictly positive functions on .
Given and (see [4]), we obtain that for ,
If , then Equation (63) holds for as well.
For and , the -mixed volume is defined by the following:
From Equations (64) and (63), we get that for (see [45]),
The -surface area of is given by .
The following definition will be required.
Definition 8 (see [4]). Let . If is a Borel measure on and , then , the image of under , is a Borel measure such that for each Borel .
Recall that the -mixed volume has a dual integral formulation (see [4]): If , then where is the radial Gauss map of .
For and , we define the volume-normalized -mixed volume by the following:
Note that is the normalized dual conical measure of , it is a probability measure on . Let . Then,
The mixed entropy of is defined by the following:
Note that . As the case in Equation (63), for the dual mixed entropy, we have that for ,
2.2. The Radial Gauss Map
The following results come from the articles [3, 4].
Suppose is a convex body in . For each , the hyperplane is called the supporting hyperplane to with outer normal vector .
The spherical image of is defined by the following:
The reverse spherical image of is defined by the following:
Suppose is a set consisting of all , for which the set , which we frequently abbreviate as , contains more than a single element. It is a well-known fact that (see Schneider [46], p. 84). The function on the set of regular radial vectors of is precisely defined by the following: by making be the unique element in for each , The function is called the spherical image map of and is known to be continuous (see Schneider [40], Lemma 2.2.12). It will be very convenient to abbreviate by . Since , when the integration is about , it does not matter if the domain is over subsets of or .
The set consisting of all , for which the set contains more than a single element, is of –measure (see Schneider [40], Theorem 2.2.11). The function is precisely defined on the set of regular unit normal vectors of : by making be the unique element in , for each . The function is called the reverse spherical image map and is well known to be continuous (see Schneider [40], Lemma 2.2.12). By extending to be a homogeneous function of degree in , we get a natural definition of on the set of all regular normal vectors on .
For , the radial Gauss image of is defined by the following:
For a subset , the reverse radial Gauss image of is defined by the following:
Thus,
In particular, we can see that if contains only a single vector ,
Note that Equation (78), and hence for and , we see from Equation (77) that
Thus, for ,
We shall need to make use of the fact that for ,
If , then , and Equation (77) becomes and hence Equation (84) holds for almost all , with respect to spherical Lebesgue measure.
The following lemma will be used.
Lemma 9 (see [4]). If , then for each .
Since for almost all with respect to spherical Lebesgue measure, and for almost all with respect to spherical Lebesgue measure, Lemma 9 implies that if , then almost everywhere with respect to spherical Lebesgue measure.
For , the radial map of is defined by the following: for . Note that is just the restriction to of the map .
The radial Gauss map of the convex body is defined by the following: where . Since is a bi-Lipschitz map between the spaces and , so it follows that has spherical Lebesgue measure . We observed that if , then contains only the element . Since both and are continuous, is continuous. Notice that for , and hence for ,
If , we see that with from the definition of . Hence from Equation (89) we have and we get the following (see [4]):
Combining with Equations (86) and (91), we have the following: for almost all with respect to spherical Lebesgue measure.
The surface area measure of a convex body can be defined, for Borel , by the following: where is the reverse spherical image of .
If the boundary of a convex body , denoted by , is smooth with positive Gauss curvature, the surface area measure of is absolutely continuous with respect to spherical Lebesgue measure. The density can be regarded as the reciprocal of Gauss curvature and expressed in terms of the support function and its Hessian matrix on : where denotes the Hessian matrix of and is the identity matrix with respect to an orthonormal frame on . See Schneider [46].
For and , its -surface area measure introduced in [2] is defined by the following: or equivalently by the following: for each Borel , where is the spherical image function of .
For , we easily see and . Then, Equation (91) implies the following:
The following integral identity is established in [3].
Lemma 10. If and , while is bounded and Lebesgue integrable, then
In [3], we see that
Lemma 11. If is strictly convex, and and are both continuous, then where is the gradient of in , and is defined only on , the set has measure .
We will require a slight extension of Equation (97). To be specific, if , while is strictly convex, and and are both continuous, then (see [4])
The following lemma will be used.
Lemma 12 (see [4]). For each , the set is dense in .
3. -Curvature Measures
For a star body , define by letting (see [4])
Note that is continuous and positively homogeneous of degree . If is an origin-symmetric convex body in , then is just an ordinary norm in , and is the -dimensional Banach space whose unit ball is .
Note that the definition (Equation (102)) is an extension of Minkowski functional (Equation (53)) of convex body .
Definition 13. Suppose . For , the -th area measure is defined by the following: for each Lebesgue measurable , and the -th curvature measure is defined by the following: for each Borel . Moreover, for each , the -curvature measure is defined by the following:
Observe that
Note that from definition (Equation (104)) and the fact that Equation (84) holds off of the set of spherical Lebesgue measure , so for each Borel , we get the following:
That is,
We observed that is absolutely continuous with respect to spherical Lebesgue measure. Then, from Equation (108), we deduce that
Lemma 14. Let and . If each function is bounded and Borel, then
Proof. Because Equation (109) is shown by Equation (108) as an indicator function of the Borel set, we see that Equation (109) holds for a linear combination of the indicator functions of the Borel set, namely, simple functions , is given by the following: where and Borel . Now let us choose a sequence of simple functions converging to the bounded Borel function . Note that is bounded, can be selected as uniformly bounded. Then, converges pointwise to on . Since is a Borel function and the radial Gauss map is continuous; thus, is a Borel function on . Because is bounded, and has spherical Lebesgue measure , we can infer that is integrable, and is spherical Lebesgue integrable in . Since is a finite measure, by taking the limit , we obtain Equation (109).☐
Proposition 15. Let . If , then for each Borel set .
Proof. From Equations (105), (109), and (84), we have for each Borel , ☐
Obviously, the total measures of the -th curvature measure and the -th area measure are the -th mixed volume, i.e.,
It follows immediately from Equations (103) and (104) that
The -curvature measures have the following properties.
Property 16. Let . If . Then, for each Borel set and each bounded Borel function , we have the following:
Proof. Because is a bounded Borel function, from Equation (109) with , we have the following:
Thus, in light of Equation (105) is the desired result (Equation (115)).
By Equations (115), (89), and (90), and letting and in Equation (98), we have the following:
☐
This yields Equation (116).
Take in Equation (116). Notice that for almost all with respect to spherical Lebesgue measure. So, we immediately obtain Equation (117).
Remark 17. Equation (115) tells us the rationality for Definition 1 of the -curvature measure .
Example 18 (-curvature measures of polytopes). Suppose be a polytope with outer unit normal vectors . If is a cone consisting of all rays emanating from the origin and passing through the face of whose outer normal is . Remember that we abbreviate by , and from Equation (80), we get the following: If is a Borel set such that , then has spherical Lebesgue measure . So, the -curvature measure is discrete and concentrated on . From Proposition 15 and Equation (120), we have the following: where represents the delta measure centered on , and
Example 19 (-curvature measures of strictly convex bodies). Let are strictly convex. Suppose is continuous, then we start with Equations (116) and (100)(taking ) and combine the fact that has measure , it follows that
Using Equation (95), this shows that
Example 20 (-curvature measures of smooth convex bodies). Let has a boundary with everywhere positive Gauss curvature. Because in this case, is absolutely continuous for the spherical Lebesgue measure; therefore, is absolutely continuous for the spherical Lebesgue measure, and from Equations (124), (94), and (47), we get the following: where represents the gradient of on at and represents the Hessian matrix of with respect to an orthonormal frame on . We write Equation (125) as , that is, We say convex body with respect to a fixed convex body as a parameter have a positive continuous -curvature function .
The weak convergence of -curvature measure is an important property contained in the following propositions.
Proposition 21. Let and . If with , then , weakly.
Proof. Let is continuous. From Equation (115) we know that for all . Since , with respect to the Hausdorff metric, we have that , uniformly on , and the surface area measure has the following property (see [2, 7, 23]): Thus, It follows that , weakly. ☐
The following statement contains the absolute continuity of -curvature measure with respect to surface area measure.
Proposition 22. Let . If , then -curvature measure is absolutely continuous with respect to the surface area measure .
Proof. Let be such that , or equivalently by definition (Equation (96)), . Then, Equation (117) states that Thus, the integration is over a set of measure .☐
The following proposition shows that the -curvature measure including the classical surface area measures and the -surface area measures. Therefore, the classical surface area measures and the -surface area measures are special cases of the -curvature measures.
Proposition 23. Suppose and . Then,
Proof. Let be a Borel set. From Equations (117) and (96), we have the following:
Therefore, we get Equations (131) and (133).
From Equations (117), (54), (90), and (96), we have the following:
Therefore, we get Equation (132). Similarly, we can get the rest.☐
Recall that the concept of the valuation. A function defined on the space of convex bodies and taking values in an abelian semigroup is called a valuation if whenever .
The set of Borel measures on is represented by . We are going to prove that now, for fixed indices , and a fixed convex body , the functional , defined by is a valuation; namely, if , are such that then
To prove the valuation of -curvature measure, we shall employ Weil’s approximation lemma (see [4]):
Lemma 24. If are such that is convex, then and may be approximated by sequences of bodies that are both strictly convex and smooth and such that .
We appeal to Proposition 21 together with Weil’s approximation lemma in order to complete our proof.
Theorem 25. Suppose and . Then, the functional defined by , is a valuation.
Proof. We will use the fact that if are such that , then and . We will also take advantage of the fact that and are defined almost everywhere on the boundaries of and , respectively.
First of all, let us assume that and are both strictly convex. For a fixed , write as the union of three disjoint pieces , where
while
In this case, we have the following:
while
Alternatively, using Equation (117), this has
Similarly,
It is also the case that
In order to see the fact that the last one, we observe that the strict convexity of and forces .
Using the fact that is a measure in the third argument on , combined with the fact that the union is disjoint, by adding Equations (145), (146), and (147) we obtain that
which is the desired result.
For any , we resort to Proposition 21 in order to use the weak continuity of in the first argument.☐
4. Variational Formulas for -Mixed Volumes
Suppose is a closed subset of that is not contained in any closed hemisphere. Let and be consecutive, and . Let be a positive continuous function defined as follows: for each , where is continuous and , uniformly on . And denote by
Wulff shape determined by . We call the logarithmic Wulff shape family generated by . If is the support function of convex body , we also put written .
Let and be continuous, and . Let be a positive continuous function defined by the following: for each , where again is continuous and uniformly on . And denote by the convex hull generated by . We call the logarithmic family of convex hull generated by . If is the radial function of convex body , we also put as .
The following lemma shows that the support functions of a logarithmic family of the polar of convex hulls are differentiable with respect to the variational variable.
Lemma 26. Suppose be a closed set that is not contained in any closed hemisphere of . Let and be continuous. If is a logarithmic family of convex hulls of and , then for all ; namely, for all regular normals of , where Equation (153) holds a.e. with respect to spherical Lebesgue measure. Moreover, there exist and so that for all and all .
Proof. Obviously,
Therefore,
Since and are two convex bodies in , and as , there exist and such that
for each . From this, it follows that there exists so that
It is easily seen that whenever , whereas whenever . Thus,
It follows that
that is
on , whenever .
Let . Since as uniformly on , we may choose so that for all , we have on . From Equation (151) and the definition of , we immediately see that
on , whenever . Let . Together with Equations (161) and (162), we give Equation (154).☐
The following theorem gives variational formulas for the -mixed volume and -mixed entropy for a family of logarithmic convex hulls.
Theorem 27. Let is a closed set not contained in any closed hemisphere of . If and are continuous, and is a logarithmic family of convex hulls of , then for and , for ,
Proof. Abbreviate by . Recall that is the set of spherical Lebesgue measure zero that consists of the complement, in , of the regular normal vectors of the convex body . Note that the continuous function
is well defined by for all .
Let . To see that , let
for some . This means that
and hence . Because in addition to obviously belonging to , it also belongs to . But is a regular normal vector of , and therefore, . Then,
From this, Equation (168), Equation (52), and Lemma 9 yield the following facts:
As is closed, by using the Tietze extension theorem, extend the continuous function to a continuous function . Therefore, using Equation (169) we see that
for .
Using Equation (22), the fact that has measure zero, Equation (51), Equation (154), the dominated convergence theorem, Lemma 26, Equation (86), Equation (170), Lemma 14, and again Equation (170), we have the following:
☐
According to Equations (70) and (51), the fact that has measure zero, the dominated convergence theorem, Equation (151), together with Equations (170) and (86), Lemma 14, and again Equation (170), we have the following:
Using the same argument as in the second part of the proof, we get that
The following theorem gives the variational formulas for the -mixed volumes and mixed entropy of the logarithmic family of Wulff shapes.
Theorem 28. Suppose is a closed set not contained in any closed hemisphere of . Let and be continuous, and be a logarithmic family of Wulff shapes associated with . If , then for , for ,
Proof. The logarithmic family of Wulff shape is defined as the Wulff shape of , where is given by the following: Let . Then, Let be the logarithmic family of convex hulls associated with . Then from Lemma 7, we obtain that and the desired conclusions now follow from Theorem 27.
We describe the special cases of Theorem 27 and Theorem 28 for logarithmic families of convex hull and Wulff shape generated by convex bodies.
Theorem 29. If and is continuous, then for , for ,
Proof. In Theorem 27, let . Then, . In particular, from (53) we have .☐
Above variational formulas for convex hulls imply variational formulas for Wulff shapes.
Theorem 30. If and is continuous, then for , for ,
Proof. The logarithmic family of Wulff shapes is defined by the Wulff shape , where This, and the fact that , allows us to define and will generate a logarithmic family of convex hulls . Since , Lemma 7 gives the following: Therefore, Theorem 30 now follows directly from Theorem 29.☐
The following theorem gives the variational formulas of -mixed volumes and mixed entropies with respect to Minkowski combinations.
Theorem 31. If and , then for , for and , for and , and if ,
Proof. For small , is defined by the following: From Equations (61) and (62), the Wulff shape . For sufficiently small , it follows from Equation (191) that Let when , and let when . The required formulas now follow Theorem 30 and Equation (105).☐
We use the normalized power function, and we can write the formula in Theorem 31 as a single formula.
Theorem 32. Suppose . For ,
For Minkowski linear combinations, it would help to have an affine version of Theorem 31. This is contained in
Theorem 33. Suppose . If , then
Proof. Let From Equation (58) we know the Wulff space . From the above definition of , it follows immediately that for sufficiently small , Let . The desired formulas now follow directly from Theorem 30.☐
Theorem 34. If and , then for all and ,
Proof. Obviously, the case and is handled by Equation (200). The case and is handled by Equation (201), while the case and is handled by Equation (202).
We adopt the methods and techniques of paper [4]. Recall that Haberl and Parapatits refer to the [9] classified measure-valued operators on , which are -inverse degree and corresponding to the transformation behavior in Theorem 34. From Equations (63), (65), and (186), we see that for all and all ,
or equivalently for all and all ,
By Definition 8, and note the important fact that support functions are positively homogeneous of degree , from Equations (45) and (204), we have the following:
This shows that the measures and when integrated against the -th power of support functions of bodies in are identical; thus, Lemma 12 now indicates that
it can be concluded that Equation (199).
The proof for Equation (200) is the same as the proof for Equation (199): As long as , it will be the case that Equation (204) continues to hold even if provided we appeal to Equations (188) and (71) when previously we had turned to Equations (188) and (65).
From Equations (63), (65), and (194), we know that for all and all ,
In Equation (207), choose . Then, by Equation (45), we see that , and (6.15) becomes the following form:
for all and all . Together with Equations (207) and (208), we have the following:
this and Equation (45) give that for all and all ,
Equivalently,
for all and all . Using Lemma 12, we see that Equation (211) yields
for all and all . This establishes Equation (201).
The proof of Equation (202) is identical to the proof of Equation (201) except that instead of appealing to Equations (194) and (65) we appeal to Equations (195) and (71).☐
5. The -Mixed Volumes
For , the -mixed volume has the integral representation
From Equation (115), with and , we have that
By Equation (131), the -mixed volume has a dual integral formulation. If , then
The dual integral formulation of -mixed volume was first introduced by Lutwak et al. in [4]. This leads us to define following -mixed volumes.
Definition 35. Let and . The -mixed volume is defined by the following:
Using Equation (115) with , Equation (216) can be written as follows:
From Equations (216) and (124), the -mixed volume can be written as follows: where the function .
From -mixed volume (Equation (30)) (or Equation (217)), the -mixed volume (Equation (9)) (or Equation (22)) will be shown to be the special cases.
Proposition 36. Suppose . If , then
Proof. Identity (Equations (219)–(221)) follow from Equation (22) and Equation (34) (or Equation (217)). Similarly, we can prove Equations (222) and (223).
Proposition 37. The -mixed volume is -invariant. That is, for , , and ,
Proof. For , the conclusion follows from Equation (223) and the -invariance of -mixed volumes (Equation (65)). We assume . By Definition 35, Equation (199), and Equation (200), the fact that support functions are positively homogeneous of degree , Equation (45), and Definition 8, we have the following: From the dual Equation (217) of -mixed volume and Equation (44), we have for real ,
Proposition 37, together with Equations (216) and (226), shows that for ,
For -mixed volume, the following inequality is a generalization of the -Minkowski inequality for -mixed volume.
Theorem 38. Suppose are such that and . If , then with equality if and only if are dilates when and are homothetic when .
Proof. From Equations (21) and (217), we have the following: From this, by the Hölder inequality (see [47]), the dual integral formulation (Equation (22)) of -mixed volume and -Minkowski inequality (Equation (11)), we have the following:
The equality conditions follow from the equality conditions of Hölder inequality and the -Minkowski inequality (Equation (11)) for -mixed volumes. Namely, the equality for the above inequality holds if and only if are dilates when and are homothetic when .
Over the past three decades, valuation theory has become an ever more important part of convex body geometry. See, e.g., [11–13, 18, 48–53]. The convex -mixed volume is the valuation for each entry.
Proposition 39. The -mixed volume is a valuation over with respect to all , and .
Proof. The -mixed volume is a valuation on respect to the third argument can be seen easily by writing Equation (216) as follows:
and from Equation (139) (or Theorem 25), observing that for , we have the following:
Together with Equations (216) and (232), we have the following:
Namely, is a valuation in the third argument.
Observing that for such that . Then, we have the following:
Note that and . Together with Equations (216) and (234), we see that is a valuation in the second argument, i.e,
Note that if are such that , then we have the following:
Together with Equations (218) and (236), we see that is a valuation in the first argument, i.e,
Let . The -th mixed cone-volume measure of and is a Borel measure on the unit sphere is defined by for a Borel and ,
Since the -th mixed volume, has a dual integral formulation:
We can turn the -th mixed cone-volume measure into the probability measure on the unit sphere by normalizing it by -th mixed volume of the bodies. The -th mixed cone-volume probability measure of and is defined by the following:
If , then for each real , we define the normalized -mixed volume by the following:
Let . We give the following:
The -th mixed entropy of convex bodies is defined by the following:
In particular,
☐
6. The -Minkowski Problems
The existence and uniqueness of -Minkowski problem is the central problem to be investigated here. Its existence problem can be expressed as follows:
Problem 40. Let , and is fixed. Given a Borel measure , what are necessary and sufficient conditions on such that there exists a whose -curvature measures is the given measure ?
-Minkowski problem when . When the given data measure has a density , it follows from Equation (125) that -Minkowski problem is equivalent to solving the following Monge-Ampère-type equation on : where is the unknown function on , and is the gradient vector function in of the extension from to as a vector function that is positively homogeneous of degree . If we assume that the range of the gradient function is , then is also an unknown function related to .
Our uniqueness result for the -Minkowski problem is presented in the following:
Problem 41. For fixed and , if such that then how is related to ?
Now, we establish uniqueness of the solution to the problem with for the case of polytopes.
Theorem 42. Let be polytopes and let . Suppose
Then, when and is a dilate of when .
Proof. According to Equations (121) and (122), we get that the curvature measures of polytopes are discrete, and that implies that and must have the same outer unit normal vectors and
where denotes the delta measure concentrated at , and
Here and are the cones formed by the origin and the facets of and with vector , respectively.
Assume that . Tt is easy to see that is not possible. Set be the maximal number with . This has . Since and have the same outer unit normal vectors, there is a facet of which is contained in a facet of . The outer unit normal vector of those facets is denoted by . It follows that
Thus,
with equality if and only if . By this and Equation (249), we can obtain that
But implies that if . Obviously, this is a contradiction.☐
If , then Equation (249) forces equality in Equation (251). So, , and the facets of and with outer unit normal vector are the same. Let is the outer unit normal vector to a facet, which is adjacent to the facet whose outer unit normal vector is . Thus, the facet of with outer unit normal vector is contained in the facet of with outer unit normal vector . A similar argument holds that the two facets are the same. Continuing in this manner, it follows that .
7. Several Other Problems
Here, we present several issues that need to be discussed in the future. Some of the definitions and problems below are different from the paper [40, 43, 44, 54].
7.1. -Mixed Affine Surface Areas
In [7], Lutwak defined the -affine surface area for by the following:
Hug in [55] observed that the -affine surface area is well defined for .
The following affine isoperimetric inequality was established in [7] for , and in [56] for . If , then with equality if and only if is an ellipsoid. Here, is the volume of the dimensional unit sphere.
Definition 43. Suppose . For and , the -th curvature measure of (related to star body ) is defined by the following: for each Borel , and -curvature measure of is defined by the following: for each Borel .
It follows from Definition 43 that
Definition 44. Suppose . If , the -th mixed volume is defined by the following:
Definition 45. Suppose . If and , the -mixed volume of and (with respect to ) is defined by the following:
Inspired by [40, 54], from Equations (258) and (259) we define -mixed affine surface area as follows:
Definition 46. For and , the -mixed affine surface area of (relate to ) is defined by the following:
When , from Equation (219) we have the following:
is the -affine surface area .
Problem 47. For the -mixed affine surface area, does it maintain affine invariance and continuity? How to establish its affine isoperimetric inequality?
7.2. -Mixed Geominimal Surface Area
In [7], Lutwak defined the -geominimal surface area by the following:
and proved the following affine isoperimetric inequality: If , then with equality if and only if is an ellipsoid.
Motivated by the -mixed geominimal surface area (Equation (257)), we define -mixed geominimal surface area, , of relative to as follows:
Definition 48. For , and , the -mixed geominimal surface area of relative to is defined by the following:
When , from Equation (219) we have the following:
is the -geominimal surface area .
Problem 49. For the -mixed geominimal surface area, does it maintain affine invariance and continuity? How to establish its affine isoperimetric inequality?
7.3. -John Ellipsoids
Suppose and is a convex body in with the origin in its interior. Among all origin-symmetric ellipsoids , the unique ellipsoid that solves the constrained maximization problem:
is called the -John ellipsoid of which defined in [45] and denoted by . Clearly, . Here, is the normalized -mixed volume of and . In the case , we define the following:
In general, the -John ellipsoid is not contained in (except when ). However, when , it has . In reverse, for , the version of ball’s volume-ratio inequality [45] states that with equality if and only if is a parallelotope.
We know that from Equation (241), for , the normalized -mixed volume is calculated by the following:
In the case , define the following:
By Equation (271), we have the following:
Let denote the class of origin-symmetric ellipsoids in . Inspired by the constrained maximization problem (Equation (266)), the reader may consider its -version.
Problem 50. Let . For , find an ellipsoid, among all origin-symmetric ellipsoids, which solves the following constrained maximization problem: An ellipsoid that solves the constrained maximization problem will be called -John ellipsoid for and denoted by .
In particular, when , from Equations (219) and (22), we have the following:
Thus, . So, Problem 50 degenerates into the problem.
Data Availability
All data included in this study are available upon request by contact with the corresponding author.
Conflicts of Interest
The author declares that there are no competing interests.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (11561020). The author is particularly grateful to Professor Weidong Wang, Dr. Yibin Feng, and Dr. Denghui Wu for their comments on various drafts of this work.