Issue 
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 4, August 2022



Page(s)  281  286  
DOI  https://doi.org/10.1051/wujns/2022274281  
Published online  26 September 2022 
Mathematics
CLC number: O 193
Some Inequalities about the General L_{p}Mixed WidthIntegral of Convex Bodies
^{1}
College of Science, China Three Gorges University, Yichang 443002, Hubei, China
^{2}
College of Science & Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, Hubei, China
^{†} To whom correspondence should be addressed. Email: ywj001001@163.com
Received:
10
April
2022
The BrunnMinkowski type and the cyclic BrunnMinkowski type inequalities for the ith general L_{p}mixed widthintegral of convex bodies are established. Further, two cyclic inequalities for the differences of ith general L_{p}mixed widthintegral of convex bodies are obtained.
Key words: BrunnMinkowski type inequality / cyclic inequality / general L_{p}mixed widthintegral
Biography: MA Jianyi,male,Master, research direction:convex geometric analysis. Email:1347599284@qq.com
Fundation item: Supported by the National Natural Science Foundation of China (11901346)
© Wuhan University 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 Introduction and Main Results
The setting for this paper is dimensional Euclidean spaces . Let denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean space . denotes the set of convex bodies containing the origin in their interiors. denotes the ndimensional volume of a body , the standard unit ball, and . Let denote the unit sphere in .
Blaschke^{[1] }considered the classical widthintegral of convex bodies first and Hadwiger^{[2]} studied it further. In 1975 Lutwak^{[3]} introduced the ith widthintegral of convex bodies. In 1977, Lutwak^{[4]} generalized the ith widthintegral to the mixed widthintegral of convex bodies. In 2016, Feng^{[5]} gave the definitions of mixed widthintegral and the general ith widthintegral of convex bodies. In 2017, Zhou^{[6] }defined the general L_{p}mixed widthintegral of convex bodies. For the more results of the mixed widthintegral of convex bodies, we refer the interested reader to Refs. [714].
In this paper, we first establish the BrunnMinkowski type inequality for the ith general L_{p}mixed widthintegral of convex bodies.
Theorem 1 Let , , , and have similar general L_{p}width,for , , or , , , we have
for , , we have
the equality holds in (1) or (2) if and only if and have similar general L_{p}width, and , where is a constant.
We also establish two cyclic inequalities for the differences of ith general L_{p}mixed widthintegral of convex bodies.
Theorem 2 Let , , , , , has constant general L_{p}width, for , , we get
with equality if and only if has constant general L_{p}width.
Theorem 3 Let , , , , , and have similar general L_{p}width, for , , we can obtain
with equality if and only if , where is a constant, and and have similar general L_{p} width.
Meanwhile, we establish a cyclic BrunnMinkowski inequality for the ith general L_{p}mixed widthintegral of convex bodies.
Theorem 4 Let ,, , , if and , we can obtain
with equality if and only if and both have constant general L_{p}width. If and , or and , the inequality is reversed.
1 Preliminaries
1.1 Support Function and Firey L_{p}Combination
If , the support function, , is defined by^{[15,16]}
where denotes the standard inner product of and .
For , and , (not both zero), the Firey L_{p}combination of and is defined by^{ [17]}
where the operation "" is called Firey addition and denotes the Firey scalar multiplication.
1.2 General L_{p}Mixed WidthIntegral of Order i
For and , the general L_{p}mixed widthintegral of is defined by ^{[6]}
where for any , and , are chosen as follows:
and satisfy
and are said to have similar general L_{p}width if there exists a constant such that for all . If for all , then we call and have the same general L_{p}width. If is a constant for all , we call has the constant general L_{p}width.
Taking and in (7), the general L_{p}mixed widthintegral of is given by
Further, let in (8), since , and write for , we get
where is called the ith general L_{p}mixed widthintegral of . If , we write , is called the general L_{p}widthintegral of .
2 Proofs of Theorems
In this section, we give the proofs of the Theorems 14. The proof of Theorem 1 requires the following lemmas.
Lemma 1^{[6]}If , and , for and , then
for and , then
with equality in every inequality if and only if and have similar general L_{p}width.
Lemma 2^{[18,19]}Let and be two series of nonnegative real numbers, and , , for , then
for or ,
with equality in every inequality if and only if and are proportional.
Proof of Theorem 1 Let in Lemma 1, since is a constant, for , we immediately obtain
and for ,
the equality holds if and only if and have similar general L_{p}width, which is just Corollary 3.
Since ,,,, and have similar general L_{p}width, for , we have
By the definition of the ith general L_{p}mixed widthintegral of convex bodies, we know that and if, for ; and if , for . According to (13), combining (16) with (17), we can obtain
the equality holds if and only if is proportional to , and and have similar general L_{p}width. The proof of inequality (2) is similar. This proves the theorem.
The proof of Theorem 23 requires the following lemma.
Lemma 3^{[20]}Suppose that , (= 1,2) are nonnegative continuous functions on such that and for , , and for all , where is a constant, then
with equality if and only if for any .
Proof of Theorem 2 Suppose that , let
in Lemma 3, since , has constant general L_{p}width, /= is a constant, according to Lemma 3, we can get the inequality (3).
By Lemma 3, the equality in (3) holds if and only if for all , that means has constant general L_{p}width. This proves the theorem.
Proof of Theorem 3 We can prove Theorem 3 by Lemma 3 as well. Suppose that and , let
in Lemma 3, since , , and have similar general L_{p}width, is a constant. We get the consequence.
By Lemma 3, the equality in (4) holds if and only if for all , that means and have similar general L_{p}width. This proves the theorem.
Taking , , in Theorem 3, we obtain
Corollary 1 Let ,,,, , , and have similar general L_{p}width, for , , one gets
with equality if and only if , where is a constant.
Proof of Theorem 4 If , combined (6) with (9), according to the Minkowski's inequality^{[21]}, it follows that
Since means , using Hölder's inequality^{ [22]} we have
Hence, we can get the following inequality
Similarly, we can also obtain
By (21), (23) and (24), we get
From the equality condition of the Minkowski's inequality, we see that the equality (21) holds if and only if and have similar general L_{p}width. By the equality conditions of Hölder's inequality, equality holds in (22) if and only if has constant general L_{p}width. Similary, the equality holds in (24) if and only if has constant general L_{p}width. Thus, the equality holds in (5) or its reverse if and only if both and have constant general L_{p}width.
In particular, take in Theorem 4. Since K +{o}=K, and notice that by inequality (5), we can obtain the following Corollary.
Corollary 2 Let , , , for , then
with equality if and only if has constant general L_{p}width.
Let in Theorem 4, we may obtain the following BrunnMinkowski inequality for the ith general L_{p}mixed widthintegral.
Corollary 3 For , , , , if , then
with equality if and only if and have simliar general L_{p}width. If , or , inequality (27) is reversed.
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