The -dual mixed geominimal surface area for multiple star bodies
© Li and Wang; licensee Springer. 2014
Received: 27 August 2014
Accepted: 4 November 2014
Published: 13 November 2014
According to the notion of the -mixed geominimal surface area of multiple convex bodies which were introduced by Ye et al., we define the concept of the -dual mixed geominimal surface area for multiple star bodies, and we establish several inequalities related to this concept.
Let denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean space . For the set of convex bodies containing the origin in their interiors and the set of convex bodies whose centroids lie at the origin in , we write and , respectively. and , respectively, denote the set of star bodies (about the origin) and the set of star bodies whose centroids lie at the origin in . Let denote the set of that have a positive continuous curvate function. Let denote the unit sphere in and the n-dimensional volume of the body K. For the standard unit ball B in , its volume is written by .
Here denotes the polar body of Q, and denotes a type of -mixed volume of , (see ).
Here denotes the -dual mixed volume of (see Section 2).
Note that we extend L from an origin-symmetric convex body to in definition (1.1). Actually, we can prove that the results of  all are correct under this extension.
In this paper, we first define the -dual mixed geominimal surface area for multiple star bodies with the same idea in mind as .
Here denotes a type of -dual mixed volume of the star bodies , (see (2.5)).
Further, we establish some inequalities for the -dual mixed geominimal surface area. Our results can be stated as follows.
In particular, if , then we have the following.
Equality holds in the second inequality of (1.8) if and only if () all are dilates of each other.
Using Corollary 1.1, we may get the following Blaschke-Santalö type inequality.
Equality holds in the second inequality of (1.9) if and only if () all are balls centered at the origin.
Equality holds in (1.10) and (1.11) if and only if each ().
2 Notations and background materials
2.1 Radial function and polar set
If is positive and continuous, K will be called a star body (with respect to the origin). Two star bodies K and L are said to be dilates (of one another) if is independent of .
with equality if and only if K is an ellipsoid centered at the origin.
2.2 Dual mixed volume
with equality if and only if are all dilations of each other.
with equality if and only if are all dilations of each other.
2.3 -Dual mixed volume
3 Results and proofs
In this section, we will prove Theorems 1.1-1.3 and Corollaries 1.1-1.2.
Proof of Theorem 1.1 We first prove inequality (1.7) is true.
with equality if and only if there exist constants (not all zero) such that for all .
This gives (1.7).
for all , where ().
for all , i.e., () all are dilates of each other. □
This gives (1.8).
From the equality condition of (1.6), we easily find that equality holds in the second inequality of (1.8) if and only if there exist constants (not all zero) such that, for all , . This means all () are dilates of each other. □
In order to prove Corollary 1.2, we give the following lemma.
Lemma 3.1 ()
with equality if and only if K is a ball centered at the origin.
This yields (1.9).
By the equality conditions of inequality (3.4) and the second inequality of (1.8), we know that equality holds in the second inequality of (1.9) if and only if all are balls centered at the origin. □
This gives the proof of Theorem 1.2. □
According to the equality condition in the Hölder inequality, we know that equality holds in (3.7) if and only if there exist constants () such that for any , i.e., for each , and both are dilates.
This is just (1.11). Because of each in inequality (3.7), together with the equality condition of (3.7), we see that equality holds in (1.11) if and only if each .
From the equality condition in (1.11), we see that equality holds in (1.10) if and only if each . □
The authors would like to sincerely thank the referees for very valuable and helpful comments and suggestions which made the paper more accurate and readable. Research is supported in part by the Natural Science Foundation of China (Grant No. 11371224) and Foundation of Degree Dissertation of Master of China Three Gorges University (Grant No. 2014PY067).
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