1. Introduction
The setting for this paper is
n-dimensional Euclidean spaces
. Let
K and
L be two convex bodies (compact, convex subsets with nonempty interiors) in
.
V denotes the volume. If
K is a compact star-shaped (about the origin) set in
, then its radial function,
, is defined by (see [
1]):
If is positive and continuous, K is called a star body (about the origin), and denotes the set of star bodies in . is the subset of containing the origin in their interiors. The unit sphere in is denoted by , and B denotes the standard unit ball in .
The classical Brunn–Minkowski inequality is (see [
2])
where + denotes vector or the Minkowski sum of two sets, i.e.,
In 2004, Leng (see [
3]) presented a new generalization of the Brunn–Minkowski inequality for the volume difference of convex bodies.
Theorem 1. Suppose that and D are compact domains, and , is a homothetic copy of D. ThenThe equality holds if and only if K and L are homothetic and where μ is a constant. Leng’s result is a major extension of the classical Brunn–Minkowski inequality and attracts more and more attention (see [
4,
5,
6]).
In 1977, Lutwak introduced the notion of a mixed width-integral of convex bodies (see [
7]), and the dual notion, mixed chord-integrals of star bodies was defined by Lu (see [
8]). Later, as a part of the asymmetric
Brunn–Minkowski theory, which has its origins in the work of Ludwig, Haberl and Schuster (see [
9,
10,
11,
12,
13]), Feng and Wang generalized the mixed chord-integrals to general mixed chord-integrals of star bodies (see [
14]). For
and
, the general mixed chord-integral
is defined by
here,
, and the functions
and
are defined as follows
In 2016, Li and Wang extended the general mixed chord-integral to the general
-mixed chord integral of star bodies (see [
15]): For
,
and
, the general
-mixed chord integral
of
is defined by
Here,
is defined by
for any
, and
and
are chosen as (see [
16])
Obviously,
and
satisfy
denotes that
K appears
times, and
L appears
i times, which is
If constants
exist such that
for all
, star bodies
are said to have a similar general
-chord. For this general
-chord integral, Li and Wang gave the following inequalities (see [
15]).
Theorem 2. If and , then for for or ,with equality in each inequality if and only if K and L have a similar general -chord. Here and in the following Theorems, denotes the -radial Minkowski combination of K and L. Theorem 3. If and , then for ,with equality if and only if all have a similar general -chord. Theorem 4. If and , then for ,with equality if and only if K and L have a similar general -chord. 2. Main Results
Inspired by Leng’s idea, this article deals with the general -chord integral of star bodies and gives some inequalities for the general -chord integral difference.
Theorem 5. Let and . If K and L have similar general -chord and , , then for and for or ,with equality in each inequality if and only if M and have a similar general -chord. Theorem 6. Let and , and . If , have similar general -chord, then for ,with equality if and only if all have a similar general -chord. Theorem 7. Let and . If K and L have similar general -chord, then for ,with equality if and only if K and L have a similar general -chord. 3. Preliminaries
For
, the radial Blaschke linear combination
and the radial Minkowski linear combination are defined by Lutwak (see [
17]), respectively:
and
In 2007, Schuster introduced the notion of radial Blaschke–Minkowski homomorphism (see [
18,
19,
20,
21,
22]) as follows.
Definition 1. A map is called a radial Blaschke–Minkowski homomorphism if it satisfies the following conditions:
- (1)
is coninuous;
- (2)
is radial Blaschke Minkowski additive, i.e., for all
- (3)
intertwines rotations, i.e., , for all and .
Here, denotes the radial sum of and , and is the radial Blaschke sum of the star bodies K and L.
In 2011, Wang et al. (see [
23]) extended the notion of radial Blaschke–Minkowski homomorphism to
-radial Minkowski homomorphism as follows.
Definition 2. A map is called an -radial Minkowski homomorphism if it satisfies the following conditions:
- (1)
is coninuous;
- (2)
is radial Minkowski additive, i.e., for all
- (3)
intertwines rotations, i.e., , for all and .
Here,
denotes the
radial sum of
and
, i.e., (see [
9,
24])
For
, the
-radial Blaschke linear combination
was defined by Wang (see [
25]):
From Equations (2c) and (2d), we easily obtain
Here, we recall a special
-radial Minkowski homomorphism. In 2007, Yu, Wu and Leng (see [
26]) introduced the quasi-
intersection body
of a star body. Let
K be a star body in
, then the quasi-
intersection body
of
K is defined by:
Further, Wang (see [
23]) proved that the operator
has the following properties:
is continuous with respect to radial metric;
for all
intertwines rotations, i.e.,
, for all
and
, which means that the operator
is a special
-radial Minkowski homomorphism.
Now, we list three Lemmas useful in the proof of Theorems 5–7.
In 1997, Losonczi and Páles (see [
27]) extended Bellman’s inequality as follows:
Lemma 1. Let and be two sequences of positive real numbers and such that and . ThenIf or , thenwith equality if and only if , where v is a constant. Lemma 2 ([
28], p.26)
. If , thenwith equality if and only if Lemma 3 ([
5])
. Suppose that are non-negative continuous functions on such thatfor , andwhere λ is a constant. Thenwith equality if and only if for any 4. Proofs of Main Results
In this section, we prove Theorems 5–7.
Proof of Theorem 5. We only prove Equation (1f). The proof of Equation (1g) is similar to Equation (1f). Let
. Since
K and
L have similar general
-chord, by Equation (1b),
for
M and
,
Let
and
, then from Equations (3a) and (3b) and Lemma 1, we have
This gives the desired inequality of Equation (1f) and according to the equality condition of Lemma 1, we obtain that equality holds if and only if
M and
have a similar general
-chord. □
Notice that from the notion of -radial Minkowski homomorphism and Equation (2e), we have the following direct Corollary 1.
Corollary 1. Let and . is a radial Blaschke–Minkowski homomorphism. K and L have a similar general -chord and , , then for and for or ,with equality in each inequality if and only if M and have a similar general -chord. Further, since the intersection map is a special -radial Minkowski homomorphism, we have the following corollary
Corollary 2. Let and . If K and L have a similar general -chord and , , then for and for or ,with equality in each inequality if and only if M and have a similar general -chord. Proof of Theorem 6. Since
have a similar general
-chord, from (1d) we have for
,
For
,
The condition
means that
. From Equations (3c) and (3d) and Lemma 2, we obtain
Let
and
in Lemma 2. Then by Equation (2g)
which implies that Equation (1h) is proved. According to the equality condition of Lemma 2, we know that equality holds in Equation (1h) if and only if
all have a similar general
-chord. □
Proof of Theorem 7. For
, let
. Then,
and
. Let
and
After a simple calculation, we obtain
The left-hand side of Equation (2h) leads to
By Lemma 3, Equation (1i) immediately holds.
The equality condition of Equation (2h) means that is a constant, that is, K and L have a similar general -chord. This completes the proof. □
5. Conclusions
The asymmetric operators belong to a new and rapidly evolving asymmetric
-Brunn–Minkowski theory that has its origins in the work of Ludwig, Haberl and Schuster (see [
9,
11,
12,
16,
18,
19,
20]). The general
-mixed chord integral difference of star bodies was motivated by the notion of mixed width-integrals of convex bodies. We hope that besides the inequalities mentioned in this article, we can deduce some other inequalities in the future.