Open Access
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

© Wuhan University 2022

Licence Creative CommonsThis 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 Eucli-dean spaces . Let denote the set of convex bodies (compact, convex subsets with non-empty interiors) in Euclidean space . denotes the set of convex bodies containing the origin in their interiors. denotes the n-dimensional volume of a body , the standard unit ball, and . Let denote the unit sphere in .

Blaschke[1] considered the classical width-integral of convex bodies first and Hadwiger[2] studied it further. In 1975 Lutwak[3] introduced the i-th width-integral of convex bodies. In 1977, Lutwak[4] generalized the i-th width-integral to the mixed width-integral of convex bodies. In 2016, Feng[5] gave the definitions of mixed width-integral and the general i-th width-integral of convex bodies. In 2017, Zhou[6] defined the general Lp-mixed width-integral of convex bodies. For the more results of the mixed width-integral of convex bodies, we refer the interested reader to Refs. [7-14].

In this paper, we first establish the Brunn-Minkowski type inequality for the i-th general Lp-mixed width-integral of convex bodies.

Theorem 1   Let , , , and have similar general Lp-width,for , , or , , , we have

(1)

for , , we have

(2)

the equality holds in (1) or (2) if and only if and have similar general Lp-width, and , where is a constant.

We also establish two cyclic inequalities for the differences of i-th general Lp-mixed width-integral of convex bodies.

Theorem 2   Let , , , , , has constant general Lp-width, for , , we get

(3)

with equality if and only if has constant general Lp-width.

Theorem 3   Let , , , , , and have similar general Lp-width, for , , we can obtain

(4)

with equality if and only if , where is a constant, and and have similar general Lp -width.

Meanwhile, we establish a cyclic Brunn-Minkowski inequality for the i-th general Lp-mixed width-integral of convex bodies.

Theorem 4   Let ,, , , if and , we can obtain

(5)

with equality if and only if and both have constant general Lp-width. If and , or and , the inequality is reversed.

1 Preliminaries

1.1 Support Function and Firey Lp-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 Lp-combination of and is defined by [17]

(6)

where the operation "" is called Firey addition and denotes the Firey scalar multiplication.

1.2 General Lp-Mixed Width-Integral of Order i

For and , the general Lp-mixed width-integral of is defined by [6]

(7)

where for any , and , are chosen as follows:

and satisfy

and are said to have similar general Lp-width if there exists a constant such that for all . If for all , then we call and have the same general Lp-width. If is a constant for all , we call has the constant general Lp-width.

Taking and in (7), the general Lp-mixed width-integral of is given by

(8)

Further, let in (8), since , and write for , we get

(9)

where is called the i-th general Lp-mixed width-integral of . If , we write , is called the general Lp-width-integral of .

2 Proofs of Theorems

In this section, we give the proofs of the Theorems 1-4. The proof of Theorem 1 requires the following lemmas.

Lemma 1[6]If , and , for and , then

(10)

for and , then

(11)

with equality in every inequality if and only if and have similar general Lp-width.

Lemma 2[18,19]Let and be two series of non-negative real numbers, and , , for , then

(12)

for or ,

(13)

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

(14)

and for ,

(15)

the equality holds if and only if and have similar general Lp-width, which is just Corollary 3.

Since ,,,, and have similar general Lp-width, for , we have

(16)

(17)

By the definition of the i-th general Lp-mixed width-integral of convex bodies, we know that and if, for ; and if , for . According to (13), combining (16) with (17), we can obtain

(18)

the equality holds if and only if is proportional to , and and have similar general Lp-width. The proof of inequality (2) is similar. This proves the theorem.

The proof of Theorem 2-3 requires the following lemma.

Lemma 3[20]Suppose that , (= 1,2) are non-negative continuous functions on such that and for , , and for all , where is a constant, then

(19)

with equality if and only if for any .

Proof of Theorem 2   Suppose that , let

in Lemma 3, since , has constant general Lp-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 Lp-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 Lp-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 Lp-width. This proves the theorem.

Taking , , in Theorem 3, we obtain

Corollary 1   Let ,,,, , , and have similar general Lp-width, for , , one gets

(20)

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

(21)

Since means , using Hölder's inequality [22] we have

(22)

Hence, we can get the following inequality

(23)

Similarly, we can also obtain

(24)

By (21), (23) and (24), we get

(25)

From the equality condition of the Minkowski's inequality, we see that the equality (21) holds if and only if and have similar general Lp-width. By the equality conditions of Hölder's inequality, equality holds in (22) if and only if has constant general Lp-width. Similary, the equality holds in (24) if and only if has constant general Lp-width. Thus, the equality holds in (5) or its reverse if and only if both and have constant general Lp-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

(26)

with equality if and only if has constant general Lp-width.

Let in Theorem 4, we may obtain the following Brunn-Minkowski inequality for the i-th general Lp-mixed width-integral.

Corollary 3   For , , , , if , then

(27)

with equality if and only if and have simliar general Lp-width. If , or , inequality (27) is reversed.

References

  1. Blaschke W. Vorlesungen über Integral Geometric I, II [M]. New York: Chelsea, 1949. [Google Scholar]
  2. Hadwiger H. Vorlesungen Über Inhalt, Oberflӓche und Isoperimetrie [M]. Berlin: Springer-Verlag, 1957. [Google Scholar]
  3. Lutwak E. Width-integrals of convex bodies [J]. Proc Amer Math Soc, 1975, 53(2): 435-439. [CrossRef] [MathSciNet] [Google Scholar]
  4. Lutwak E. Mixed width-integrals of convex bodies [J]. Israel J Math, 1977, 28(3): 249-253. [CrossRef] [MathSciNet] [Google Scholar]
  5. Feng Y B. General mixed width-integral of convex bodies [J]. Journal of Nonlinear Sciences and Applications, 2016, 9(6): 4226-4234. [CrossRef] [Google Scholar]
  6. Zhou Y P. General Formula -mixed width-integral of convex bodies and related inequalities [J]. Journal of Nonlinear Sciences and Applications, 2017, 10(8): 4372-4380. [CrossRef] [Google Scholar]
  7. Li X Y, Zhao C J. On the Formula -mixed affine surface area [J]. Math Inequal Appl, 2014, 17: 443-450. [MathSciNet] [Google Scholar]
  8. Lutwak E, Yang D, Zhang G Y. Formula -affine isoperimetric inequalities [J].Journal of Differential Geom, 2000, 56(1): 111-132. [MathSciNet] [Google Scholar]
  9. Wang W D, Ma T Y. Asymmetric Formula -difference bodies [J]. Proc Amer Math Soc, 2014, 142(7): 2517-2527. [CrossRef] [MathSciNet] [Google Scholar]
  10. Wang W D, Feng Y B. A general Formula -version of Petty's affine projection inequality [J]. Taiwan J Math, 2013, 17(2): 517-528. [Google Scholar]
  11. Wang W D, Wan X Y. Shephard type problems for general Formula -projection bodies [J]. Taiwan J Math, 2012, 16(5): 1749-1762. [Google Scholar]
  12. Leng G S, Zhao C J, He B W, et al. Inequalities for polars of mixed projection bodies [J]. Sci China Ser A, 2004, 47(2): 175-186(Ch). [CrossRef] [Google Scholar]
  13. Lutwak E. Dual mixed volumes [J]. Pacific J Math, 1975, 58(2): 531-538. [CrossRef] [MathSciNet] [Google Scholar]
  14. Lutwak E. Inequalities for mixed projection bodies [J]. Trans Amer Math Soc, 1993, 339(2): 901-916. [CrossRef] [MathSciNet] [Google Scholar]
  15. Gardner R J. Geometric Tomography [M]. 2nd Edition. Cambridge: Encyclopedia Math Appl Cambridge University Press, 2006. [CrossRef] [Google Scholar]
  16. Schneider R. Convex Bodies: The Brunn-Minkowski Theory [M]. 2nd Edition. Cambridge: Encyclopedia Math Appl Cambridge University Press, 2014. [Google Scholar]
  17. Lutwak E. The Brunn-Minkowski-Firey theory I: Mixed volumes and the Minkowski problem [J]. Differential Geom, 1993, 38(1): 131-150. [MathSciNet] [Google Scholar]
  18. Beckenbach E F, Bellman R. Inequalities [M]. 2nd Edition. Berlin: Springer-Verlag , 1965. [CrossRef] [Google Scholar]
  19. Losonczi L, Páles Z S. Inequalities for indefinite forms [J]. Journal of Math Anal Appl, 1997, 205(1): 148-156. [CrossRef] [MathSciNet] [Google Scholar]
  20. Lv S J. Dual Brunn-Minkowski inequality for volume differences [J]. Geom Dedicata, 2010, 145(1): 169-180. [CrossRef] [MathSciNet] [Google Scholar]
  21. Hardy G H, Littlewood J E, Pölya G. Inequalities [M]. New York: Cambridge University Press, 1934. [Google Scholar]
  22. Bechenbach E F, Bellman R. Inequalities [M]. Berlin: Springer-Verlag , 1961. [CrossRef] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.