In this article, we obtain some results about the mean curvature integrals of the parallel body of a convex set in R^n. These mean curvature integrals are generalizations of the Santalo's results.
Cavitation bubbles behind a convex body were experimentally studied by a high speed camera and a hydrophone synchronously. The experiments were conducted in a circulating water tunnel with five various contraction rat...Cavitation bubbles behind a convex body were experimentally studied by a high speed camera and a hydrophone synchronously. The experiments were conducted in a circulating water tunnel with five various contraction ratios: β = 0.497, β= 0.6, β= 0.697, β= 0.751, and β= 0.799. The distributions of the cavitation bubble collapse positions behind the five different convex bodies were obtained by combining the images taken by the high speed camera. According to the collapse positions, it was found that no cavitation bubble was collapsed in the region near the wall until the ratio of the water head loss over the convex body height was larger than 20, which can be used to predict if the cavitation damage would occur in the tunnel with orifice energy dissipaters.展开更多
Classical problems in integral geometry and geometric probability involve the kinematic measure of congruent segments of fixed length within a convex body in R3. We give this measure from rotational formulae; that is,...Classical problems in integral geometry and geometric probability involve the kinematic measure of congruent segments of fixed length within a convex body in R3. We give this measure from rotational formulae; that is, from isotropic plane sections through a fixed point. From this result we also obtain a new rotational formula for the volume of a convex body; which is proved to be equivalent to the wedge formula for the volume.展开更多
Let K be a 1-unconditional convex bodies in Euclidean spaces.We study the asymptotic properties of two affine invariants m2(K) and S2(K) for a random simplex inside K.As an application,we discuss the asymptotic pr...Let K be a 1-unconditional convex bodies in Euclidean spaces.We study the asymptotic properties of two affine invariants m2(K) and S2(K) for a random simplex inside K.As an application,we discuss the asymptotic properties of two affine invariants m2(Bpn ) and S2(Bpn ),where Bpn = {x ∈ Rn : ‖x‖ p 1}.展开更多
This paper provides a method using fixed-point theory for the reconstruction of the triangle inscribed in convex bodies from X-ray functions in three arbitrary mutual directions.
In this paper, the relations between inclusion measures of different bodies related to convex body K and the inclusion measure of convex body K itself were obtained.
For the affine distance d(C,D) between two convex bodies C, D(?) Rn, which reduces to the Banach-Mazur distance for symmetric convex bodies, the bounds of d(C, D) have been studied for many years. Some well known esti...For the affine distance d(C,D) between two convex bodies C, D(?) Rn, which reduces to the Banach-Mazur distance for symmetric convex bodies, the bounds of d(C, D) have been studied for many years. Some well known estimates for the upper-bounds are as follows: F. John proved d(C, D) < n1/2 if one is an ellipsoid and another is symmetric, d(C, D) < n if both are symmetric, and from F. John's result and d(C1,C2) < d(C1,C3)d(C2,C3) one has d(C,D) < n2 for general convex bodies; M. Lassak proved d(C, D) < (2n - 1) if one of them is symmetric. In this paper we get an estimate which includes all the results above as special cases and refines some of them in terms of measures of asymmetry for convex bodies.展开更多
Properties of the p-measures of asymmetry and the corresponding affine equivariant p-critical points, defined recently by the second author, for convex bodies are discussed in this article. In particular, the continui...Properties of the p-measures of asymmetry and the corresponding affine equivariant p-critical points, defined recently by the second author, for convex bodies are discussed in this article. In particular, the continuity of p-critical points with respect to p on (1, +∞) is confirmed, and the connections between general p-critical points and the Minkowski-critical points (∞-critical points) are investigated. The behavior of p-critical points of convex bodies approximating a convex bodies is studied as well.展开更多
For convex bodies, the Firey linear combinations were introduced and studied in several papers. In this paper the mean width of the Firey linear combinations of convex bodies is studied, and the lower bound of the mea...For convex bodies, the Firey linear combinations were introduced and studied in several papers. In this paper the mean width of the Firey linear combinations of convex bodies is studied, and the lower bound of the mean width of the Firey linear combinations of convex body and its polar body is given.展开更多
In this paper, by using the Lp-Brunn-Minkowski theory and its dual theory, L2-version on the conjectured projection inequality is investigated, the (reverse) inclusive relationship between L2-projection body and the...In this paper, by using the Lp-Brunn-Minkowski theory and its dual theory, L2-version on the conjectured projection inequality is investigated, the (reverse) inclusive relationship between L2-projection body and the classical projection body are established, and a constrained minimization problem is solved.展开更多
In this paper, by means of the dual Brunn-Minkowski theories and methods as well as the integral transform, we have established a stability version of nonsymmetric convex bodies from intersection bodies. In addition, ...In this paper, by means of the dual Brunn-Minkowski theories and methods as well as the integral transform, we have established a stability version of nonsymmetric convex bodies from intersection bodies. In addition, a stability results of symmetric convex bodies from L_p-counterparts is established.展开更多
In this paper, we prove that any polygon P in R^2 containing a fixed smooth, strictly convex and origin-symmetric body Γ whose boundary is real analytic in its interior, can be determined by its Γ-section functions ...In this paper, we prove that any polygon P in R^2 containing a fixed smooth, strictly convex and origin-symmetric body Γ whose boundary is real analytic in its interior, can be determined by its Γ-section functions among the polygons.展开更多
In this article, some kinematic formulas for dual quermassintegral of star bodies and for chord power integrals of convex bodies are established by using dual mixed volumes. These formulas are the extensions of the fu...In this article, some kinematic formulas for dual quermassintegral of star bodies and for chord power integrals of convex bodies are established by using dual mixed volumes. These formulas are the extensions of the fundamental kinematic formula involving quermassintegral to the case of dual quermassintegral and chord power integrals.展开更多
In this paper, it is proved that the radial pth mean body RpK(p 〉 0) is homothetic to the difference body DK when K is a simplex, Furthermore, the equality Rp(Rq) = Rq(Rp) is established when p 〉 0 and q 〉 0....In this paper, it is proved that the radial pth mean body RpK(p 〉 0) is homothetic to the difference body DK when K is a simplex, Furthermore, the equality Rp(Rq) = Rq(Rp) is established when p 〉 0 and q 〉 0. It is also proved the Brunn-Minkowski inequality of radial pth mean body of simplex and uniqueness property.展开更多
In this paper, we study the characteristic properties for Lp-centroid bod- ies, and an improved version of Busemann-Petty problem for Lp-centroid bodies is obtained. In addition, using the definitions of Lp-pole curva...In this paper, we study the characteristic properties for Lp-centroid bod- ies, and an improved version of Busemann-Petty problem for Lp-centroid bodies is obtained. In addition, using the definitions of Lp-pole curvature image and Lp-affine surface area, a new proof of Busemann-Petty problem for Lp-centroid bodies is given.展开更多
We use a simple approach to estimating the Banach-Mazur distance between convex bodies and simplex.As an application of this approach,we provide a purely analytic proof for the known result sup_(C∈K^n) d_(BM)(C,△)≤...We use a simple approach to estimating the Banach-Mazur distance between convex bodies and simplex.As an application of this approach,we provide a purely analytic proof for the known result sup_(C∈K^n) d_(BM)(C,△)≤n + 2,where d_(BM)(.,.) denotes the Banach-Mazur distance,A denotes an n-dimensional simplex and K^n denotes the class of n-dimensional convex sets in R^n.展开更多
For p > 0, Lutwak, Yang and Zhang introduced the concept of L_p-polar projection body Γ_(-p)K of a convex body K in Rn. Let p ≥ 1 and K, L ? Rnbe two origin-symmetric convex bodies, we consider the question of wh...For p > 0, Lutwak, Yang and Zhang introduced the concept of L_p-polar projection body Γ_(-p)K of a convex body K in Rn. Let p ≥ 1 and K, L ? Rnbe two origin-symmetric convex bodies, we consider the question of whether Γ_(-p) K ? Γ_(-p) L implies ?_p(L) ≤ ?_p(K),where ?_p(K) denotes the L_p-affine surface area of K and K = Voln(K)^(-1/p) K. We prove a necessary and sufficient condition of an analog of the Shephard problem for the L_p-polar projection bodies.展开更多
In this paper,we establish a characterization of the polarity mapping for convex bodies for 1-dimensional convex bodies,which is a supplement to the result for such a characterization obtained by B?r?czky and Schneider.
In this paper we propose a method to construct probability measures on the space of convex bodies. For this purpose, first, we introduce the notion of thinness of a body. Then we show the existence of a measure with t...In this paper we propose a method to construct probability measures on the space of convex bodies. For this purpose, first, we introduce the notion of thinness of a body. Then we show the existence of a measure with the property that its pushforward by the thinness function is a probability measure of truncated normal distribution. Finally, we improve this method to find a measure satisfying some important properties in geometric measure theory.展开更多
基金Supported in part by NNSFC(10671159)Hong Kong Qiu Shi Science and Technologies Research Foundation
文摘In this article, we obtain some results about the mean curvature integrals of the parallel body of a convex set in R^n. These mean curvature integrals are generalizations of the Santalo's results.
基金Project supported by the Natural Science Foundation of China(Grant No.51179114)the National Basic Research Development Program of China(973 Program,2013CB035905)
文摘Cavitation bubbles behind a convex body were experimentally studied by a high speed camera and a hydrophone synchronously. The experiments were conducted in a circulating water tunnel with five various contraction ratios: β = 0.497, β= 0.6, β= 0.697, β= 0.751, and β= 0.799. The distributions of the cavitation bubble collapse positions behind the five different convex bodies were obtained by combining the images taken by the high speed camera. According to the collapse positions, it was found that no cavitation bubble was collapsed in the region near the wall until the ratio of the water head loss over the convex body height was larger than 20, which can be used to predict if the cavitation damage would occur in the tunnel with orifice energy dissipaters.
基金Supported by PROMETEO(Grant No.2010/028)UJI(Grant No.P1.1B2012-24)
文摘Classical problems in integral geometry and geometric probability involve the kinematic measure of congruent segments of fixed length within a convex body in R3. We give this measure from rotational formulae; that is, from isotropic plane sections through a fixed point. From this result we also obtain a new rotational formula for the volume of a convex body; which is proved to be equivalent to the wedge formula for the volume.
基金Project supported by the National Natural Science Foundation of China (Grant No.10671119)
文摘Let K be a 1-unconditional convex bodies in Euclidean spaces.We study the asymptotic properties of two affine invariants m2(K) and S2(K) for a random simplex inside K.As an application,we discuss the asymptotic properties of two affine invariants m2(Bpn ) and S2(Bpn ),where Bpn = {x ∈ Rn : ‖x‖ p 1}.
文摘This paper provides a method using fixed-point theory for the reconstruction of the triangle inscribed in convex bodies from X-ray functions in three arbitrary mutual directions.
基金Project supported by Youth Science Foundation of Shanghai Municipal Commission of Education( Grant No. 214511)
文摘In this paper, the relations between inclusion measures of different bodies related to convex body K and the inclusion measure of convex body K itself were obtained.
文摘For the affine distance d(C,D) between two convex bodies C, D(?) Rn, which reduces to the Banach-Mazur distance for symmetric convex bodies, the bounds of d(C, D) have been studied for many years. Some well known estimates for the upper-bounds are as follows: F. John proved d(C, D) < n1/2 if one is an ellipsoid and another is symmetric, d(C, D) < n if both are symmetric, and from F. John's result and d(C1,C2) < d(C1,C3)d(C2,C3) one has d(C,D) < n2 for general convex bodies; M. Lassak proved d(C, D) < (2n - 1) if one of them is symmetric. In this paper we get an estimate which includes all the results above as special cases and refines some of them in terms of measures of asymmetry for convex bodies.
基金The NSF(11271282)of Chinathe GIF(CXLX12 0865)of Jiangsu Province
文摘Properties of the p-measures of asymmetry and the corresponding affine equivariant p-critical points, defined recently by the second author, for convex bodies are discussed in this article. In particular, the continuity of p-critical points with respect to p on (1, +∞) is confirmed, and the connections between general p-critical points and the Minkowski-critical points (∞-critical points) are investigated. The behavior of p-critical points of convex bodies approximating a convex bodies is studied as well.
基金supported by the Youth Science Foundation of Shanghai Municipal Education Commission (Grant No.214511)the Research Grants Council of the Hong Kong SAR,China (Grant No.HKU7016/07P)
文摘For convex bodies, the Firey linear combinations were introduced and studied in several papers. In this paper the mean width of the Firey linear combinations of convex bodies is studied, and the lower bound of the mean width of the Firey linear combinations of convex body and its polar body is given.
基金supported by the National Natural Sciences Foundation of China (Grant Nos.10671117,10801140)
文摘In this paper, by using the Lp-Brunn-Minkowski theory and its dual theory, L2-version on the conjectured projection inequality is investigated, the (reverse) inclusive relationship between L2-projection body and the classical projection body are established, and a constrained minimization problem is solved.
基金Supported by the National Natural Science Foundation of China(11561020, 11371224)
文摘In this paper, by means of the dual Brunn-Minkowski theories and methods as well as the integral transform, we have established a stability version of nonsymmetric convex bodies from intersection bodies. In addition, a stability results of symmetric convex bodies from L_p-counterparts is established.
基金Project supported by the Youth Science Foundation of Shanghai Municipal Commission of Education (Grant No.214511), and in part by the Research Grants Council of the Hong Kong SAR, China (Grant No.HKU7016/07P)
文摘In this paper, we prove that any polygon P in R^2 containing a fixed smooth, strictly convex and origin-symmetric body Γ whose boundary is real analytic in its interior, can be determined by its Γ-section functions among the polygons.
文摘In this article, some kinematic formulas for dual quermassintegral of star bodies and for chord power integrals of convex bodies are established by using dual mixed volumes. These formulas are the extensions of the fundamental kinematic formula involving quermassintegral to the case of dual quermassintegral and chord power integrals.
基金Project supported by the Youth Science Foundation of Shanghai Municipal Commission of Education(Grant No.214511)the Research Grants Council of the HongKong SAR,China(Grant No.HKU7017105P)
文摘In this paper, it is proved that the radial pth mean body RpK(p 〉 0) is homothetic to the difference body DK when K is a simplex, Furthermore, the equality Rp(Rq) = Rq(Rp) is established when p 〉 0 and q 〉 0. It is also proved the Brunn-Minkowski inequality of radial pth mean body of simplex and uniqueness property.
文摘In this paper, we study the characteristic properties for Lp-centroid bod- ies, and an improved version of Busemann-Petty problem for Lp-centroid bodies is obtained. In addition, using the definitions of Lp-pole curvature image and Lp-affine surface area, a new proof of Busemann-Petty problem for Lp-centroid bodies is given.
文摘We use a simple approach to estimating the Banach-Mazur distance between convex bodies and simplex.As an application of this approach,we provide a purely analytic proof for the known result sup_(C∈K^n) d_(BM)(C,△)≤n + 2,where d_(BM)(.,.) denotes the Banach-Mazur distance,A denotes an n-dimensional simplex and K^n denotes the class of n-dimensional convex sets in R^n.
基金Supported by the National Natural Science Foundation of China(11561020,11371224)Supported by the Science and Technology Plan of the Gansu Province(145RJZG227)
文摘For p > 0, Lutwak, Yang and Zhang introduced the concept of L_p-polar projection body Γ_(-p)K of a convex body K in Rn. Let p ≥ 1 and K, L ? Rnbe two origin-symmetric convex bodies, we consider the question of whether Γ_(-p) K ? Γ_(-p) L implies ?_p(L) ≤ ?_p(K),where ?_p(K) denotes the L_p-affine surface area of K and K = Voln(K)^(-1/p) K. We prove a necessary and sufficient condition of an analog of the Shephard problem for the L_p-polar projection bodies.
基金Supported in part by the Natural Science Foundation of Hunan Province(2021JJ30235)the Scientific Research Fund of Hunan Provincial Education Department(21B0479)Postgraduate Scientific Research Innovation Project of Hunan Province(QL20220228)。
文摘In this paper,we establish a characterization of the polarity mapping for convex bodies for 1-dimensional convex bodies,which is a supplement to the result for such a characterization obtained by B?r?czky and Schneider.
文摘In this paper we propose a method to construct probability measures on the space of convex bodies. For this purpose, first, we introduce the notion of thinness of a body. Then we show the existence of a measure with the property that its pushforward by the thinness function is a probability measure of truncated normal distribution. Finally, we improve this method to find a measure satisfying some important properties in geometric measure theory.