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On mean curvature integrals of the outer parallel body of the projection of a convex body
Journal of Inequalities and Applications volume 2014, Article number: 415 (2014)
Abstract
In this paper, we obtain expressions of the mean curvature integrals of two outer parallel bodies, where the outer parallel bodies are in the distance ρ of a projection body in different space ( and ). These mean curvature integrals are the generalizations of Santaló’s results. As corollaries, we establish mean values of the mean curvature integrals and Minkowski quermassintegrals of two outer parallel bodies, respectively.
MSC:52A20, 53C65.
1 Introduction
The mean curvature integral is a basic concept in integral geometry. It connects many geometric invariants, such as area, the Euler-Poincaré characteristic, the degree of the spherical Gauss map, the Gauss-Kronecker curvature and so on. Also it has close relation to the Minkowski quermassintegral of convex body. Meanwhile, the mean curvature integral plays an important role in Chern fundamental kinematic formula. It is well known that kinematic formulas are very important and classical in integral geometry.
Under the assumptions that is the n-dimensional Euclidean space and is an r-dimensional linear subspace through a fixed point O, Santaló [1] investigated the i th mean curvature integral of a flattened convex body K in and established the expression of in terms of , where is the j th mean curvature integral of K in . On the basis of [1], Chen and Yang [2] investigated of a flattened convex body K in space forms and gave the expression of it in terms of , where is the j th mean curvature integral of K in r-dimensional geodesic submanifold, their work extends the result of Santaló in [1]. In [3], Zhou and Jiang investigated of the projection body as a flattened convex body of .
In this paper, we investigate the i th mean curvature integral of and , naturally, where and are the outer parallel bodies of in and , respectively. We give the expressions of in terms of . Besides, we obtain the mean value of . Our main results are the following theorems.
Theorem 1 Let K be a convex body with boundary ∂K in . Let be the outer parallel body of in the distance ρ in , where is the orthogonal projection of K on the r-dimensional linear subspace . Denote by () the mean curvature integrals of and by () the mean curvature integrals of in . Then:
-
(1)
If , then
(1.1) -
(2)
If , then
(1.2)where denotes the r-dimensional volume of .
Theorem 2 Let K be a convex body with boundary ∂K in . Let be the outer parallel body of in the distance ρ in , where is the orthogonal projection of K on the r-dimensional linear subspace . Denote by () the mean curvature integrals of as a flattened convex body of and by () the mean curvature integrals of as a convex body of . Then:
-
(1)
If , then
(1.3) -
(2)
If , then
(1.4) -
(3)
If , then
(1.5)where denotes the r-dimensional volume of .
Especially, letting , Theorem 2 reduces to Lemma 1 (in Section 2) proved by Santaló in 1957 (see [1, 4, 5]). In fact, the main result of [3] and Theorem 2 are similar in nature, but the coefficient in [3] is a little inappropriate. Note that the results of [1, 4, 5] play an important role in integral geometry and differential geometry and are widely used (see [3, 5–7]).
2 Preliminaries
A set in the Euclidean space is called convex if and only if it contains, with each pair of its points, the entire line segment joining them. A convex set with nonempty interior is called a convex body. The boundary ∂K of a convex body K is a convex hypersurface.
Let K be a convex body in , then ∂K is an -dimensional convex hypersurface. Assuming that ∂K is of class and P is a point of ∂K, we choose to be the principal curvature directions at the point P. Further, we suppose that are the principal curvatures at the point P, which correspond to the principal curvature directions.
Consider the Gauss map , whose differential
satisfies Rodrigues’ equations,
Then we have the mean curvature
along with the Gauss-Kronecker curvature,
The i th order mean curvature is the i th order elementary symmetric function of the principal curvatures. We denote by the i th order mean curvature normalized such that
Thus, is the mean curvature and is the Gauss-Kronecker curvature.
The i th order mean curvature integral of ∂K at P is defined by
where denotes the i th elementary symmetric function of the principal curvatures and dσ is the area element of ∂K. As a particular case, let be the area of ∂K, for completeness. Moreover, we have , where denotes the area of the -dimensional unit sphere and its value is given by the formula
For instance, if , and K is a plane convex figure in , then and . If , and K is a convex body in , then , and is the integral of mean curvature of ∂K. See [5, 7] for a detailed description.
On the other hand, we consider all the -dimensional linear subspaces through a fixed point O. Let be the orthogonal projection of K onto , denote by the volume of and by the densities of the Grassmann manifold . Then the mean value of the projected volumes is
where Grassmann manifold is the set of unoriented r-planes of through a fixed point, is the volume of given by
and
For completeness, we define
The Minkowski quermassintegral is introduced by Minkowski and is defined by
In particular, we put , .
The outer parallel body in the distance ρ of a convex figure K is the union of all solid spheres of radius ρ the centers of which are points of K. Then we have the following Steiner formula for the outer parallel body ():
As a consequence of the Steiner formula we have
Moreover, we have the relation between the mean curvature integrals of ∂K and the Minkowski quermassintegrals of K (see [4, 5, 7]), that is, the Cauchy formula
Note that the Minkowski quermassintegrals are well defined for any convex figure, whereas makes sense only if ∂K is of class .
Let K be a convex body in the r-dimensional linear subspace , and the mean curvature integrals of K as a convex surface of . Consider K as a flattened convex body of , Santaló obtained the following lemma with respect to the mean curvature integral in 1957 (see [1, 4, 5]).
Lemma 1 Let be the n-dimensional Euclidean space and be the r-dimensional linear subspace through a fixed point O in . Let K be a convex body of the dimension r in . Then K can be considered both as a convex body in and as a flattened convex body in . Then the qth mean curvature integral satisfies the conditions:
-
(1)
If , then
(2.16) -
(2)
If , then
(2.17)where denotes the r-dimensional volume of K.
-
(3)
If , then
(2.18)
Later, Jiang and Zeng [8] investigated the integral of of on the Grassmann manifold and obtained the mean value of these mean curvature integrals.
Lemma 2 Let K be a convex body with boundary ∂K in and let be the orthogonal projection of K on the r-dimensional subspace . Denote by () the mean curvature integrals of as a convex body of and by () the mean curvature integrals of K in . Then
3 Proofs of the main theorems and some corollaries
Proof of Theorem 1 We apply the Cauchy formula (2.15) to the convex body , then
Applying (2.14) to the convex body , we have
Then combining (3.1) and (3.2) gives
where in the first step we use the Cauchy formula
for flattened convex bodies.
Now, we are ready to compute the mean curvature integral of from the below three cases.
-
(1)
If , and obviously in (3.3). Then by Santaló’s result (2.16)
(3.4)
Inserting (3.4) to (3.3), we obtain
-
(2)
If , then (3.3) can be rewritten as
(3.6)
where the first equation and the last equation follow from (2.17) and (2.16), respectively.
-
(3)
If , from (2.16) and (2.17), followed by (2.18) and (3.3), then we have
(3.7)
If we take in (3.7), then
which is in fact (3.6). So combining (3.6) and (3.7) gives (1.2) and completes the proof of Theorem 1. □
Proof of Theorem 2 (1) If , and applying (2.16) and (3.3), then
-
(2)
If , then by (2.17),
(3.10)
Next, we turn our attention to the computation of the r-volume . By applying the Steiner formula to , we see that
Hence
Finally, we obtain
-
(3)
If , then by (2.18) we have
(3.14)
□
Based on Theorem 1, we begin to consider the integral of on Grassmann manifold , and obtain the following.
Theorem 3 Let K be a convex body with boundary ∂K in . Let be the outer parallel body of in the distance ρ in , where is the orthogonal projection of K on the r-dimensional linear subspace . Denote by () the mean curvature integrals of and by () the mean curvature integrals of K. Then:
-
(1)
If , then
-
(2)
If , then
Proof (1) If , by (1.1) and Lemma 2, the integral of on Grassmann manifold can be obtained as follows:
-
(2)
If , then by (1.2) and Lemma 1 we arrive at
(3.15)
Note that
and
therefore we obtain
Inserting (3.16) to (3.15) and using Lemma 2, we have
we complete the proof of Theorem 3. □
By the Cauchy formula (2.15) and Theorem 3, the following corollary can be obtained.
Corollary 1 Let K be a convex body with boundary ∂K in . Let be the outer parallel body of in the distance ρ in , where is the orthogonal projection of K on the r-dimensional linear subspace . Denote by () the Minkowski quermassintegrals of and by () the Minkowski quermassintegrals of K. Then:
-
(1)
If , then
-
(2)
If , then
Using divided by , and by Theorem 3, we immediately obtain the following corollaries.
Corollary 2 Let K be a convex body with boundary ∂K in . Let be the outer parallel body of in the distance ρ in , where is the orthogonal projection of K on the r-dimensional linear space . Denote by () the mean curvature integrals of and by () the mean curvature integrals of K. Then:
-
(1)
If ,
-
(2)
If ,
Corollary 3 Let K be a convex body with boundary ∂K in . Let be the outer parallel body of in the distance ρ in , where is the orthogonal projection of K on the r-dimensional linear space . Denote by () the Minkowski quermassintegrals of and by () the Minkowski quermassintegrals of K. Then:
-
(1)
If ,
-
(2)
If ,
References
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Acknowledgements
The authors would like to thank two anonymous referees for many helpful comments and suggestions that directly lead to the improvement of the original manuscript. The authors are supported in part by NSFC (Grant No. 11326073) and Natural Science Foundation Project of CQ CSTC (Grant No. cstc 2014jcyjA00019).
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Zeng, C., Ma, L. & Xia, Y. On mean curvature integrals of the outer parallel body of the projection of a convex body. J Inequal Appl 2014, 415 (2014). https://doi.org/10.1186/1029-242X-2014-415
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DOI: https://doi.org/10.1186/1029-242X-2014-415