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.
Let M be a compact convex hypersurface of class C2, which is assumed to bound a nonempty convex body K in the Euclidean space Rn and H be the mean curvature of M. We obtain a lower bound of the total square of mean cu...Let M be a compact convex hypersurface of class C2, which is assumed to bound a nonempty convex body K in the Euclidean space Rn and H be the mean curvature of M. We obtain a lower bound of the total square of mean curvature fM H2dA The bound is the Minkowski quermassintegral of the convex body K. The total square of mean curvature attains the lower bound when M is an （n - 1）-sphere.展开更多
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.
The lower bound for the volume of the zonotope for John-basis had been given by Ball. In this paper, a simple proof of Ball＇s inequality was first provided, then the result of Ball was generalized from John-basis to ...The lower bound for the volume of the zonotope for John-basis had been given by Ball. In this paper, a simple proof of Ball＇s inequality was first provided, then the result of Ball was generalized from John-basis to a sequence of non-zero vectors which are full rank. Furthermore, the upper bound for the volumes of zonotopes was given. Finally the inequalities were deduced for the inradius and circumradius of a certain zonotope.展开更多
The notion of mixed quermassintegrals in the classical Brunn-Minkowski theory is extended to that of Orlicz mixed quermassintegrals in the Orlicz Brunn-Minkowski theory. The analogs of the classical Cauchy- Kuhota for...The notion of mixed quermassintegrals in the classical Brunn-Minkowski theory is extended to that of Orlicz mixed quermassintegrals in the Orlicz Brunn-Minkowski theory. The analogs of the classical Cauchy- Kuhota formula, the Minkowski isoperimetric inequality and the Brunn-Minkowski inequality are established for this new Orlicz mixed quermassintegrals.展开更多
In this paper,we establish two theorems for the quermassintegrals of convex bodies,which are the generalizations of the well-known Aleksandrov's projection theorem and Loomis-Whitney's inequality,respectively....In this paper,we establish two theorems for the quermassintegrals of convex bodies,which are the generalizations of the well-known Aleksandrov's projection theorem and Loomis-Whitney's inequality,respectively.Applying these two theorems,we obtain a number of inequalities for the volumes of projections of convex bodies.Besides,we introduce the concept of the perturbation element of a convex body,and prove an extreme property of it.展开更多
A class of geometric quantities for convex bodies is introduced iu the framework of Orlicz Brunn- Minkowski theory. It is shown that these new geometric quantities are affine invariant and precisely the generalization...A class of geometric quantities for convex bodies is introduced iu the framework of Orlicz Brunn- Minkowski theory. It is shown that these new geometric quantities are affine invariant and precisely the generalizations of classical affine quermassintegrals.展开更多
In this paper,we first introduce a concept of L_p-dual Quermassintegral sum function of convex bodies and establish the polar projection Minkowski inequality and the polar projection Aleksandrov-Fenchel inequality for...In this paper,we first introduce a concept of L_p-dual Quermassintegral sum function of convex bodies and establish the polar projection Minkowski inequality and the polar projection Aleksandrov-Fenchel inequality for L_p-dual Quermassintegral sums.Moreover,by using Lutwak’s width-integral of index i,we establish the L_p-Brunn-Minkowski inequality for the polar mixed projec- tion bodies.As applications,we prove some interrelated results.展开更多
We establish the cyclic inequality for i-th L p-dual mixed volume and Lp-dual Urysohn inequality between p-mean width and Lp-dual quermassintegral. Moreover, the dual isoperimetric inequality for Lp-dual mixed volume ...We establish the cyclic inequality for i-th L p-dual mixed volume and Lp-dual Urysohn inequality between p-mean width and Lp-dual quermassintegral. Moreover, the dual isoperimetric inequality for Lp-dual mixed volume is proved, which is an extension of the classical dual isoperimetric inequality.展开更多
Based on Lutwak＇s the notion of Lp-difference bodies, Wang and Ma introduced asymmetric Lp-difference bodies and gave their extremum values for volumes. In this paper, we establish the extremum value inequalities for...Based on Lutwak＇s the notion of Lp-difference bodies, Wang and Ma introduced asymmetric Lp-difference bodies and gave their extremum values for volumes. In this paper, we establish the extremum value inequalities for the quermassintegrals and dual quermassintegrals of asymmetric Lp-difference bodies and their polars, respectively.展开更多
In this paper, we study the extremum inequalities of general L_(p)-intersection bodies. In addition, associating with the L_(q)-radial combination and Lq-harmonic Blaschke combination, we establish the Brunn-Minkowski...In this paper, we study the extremum inequalities of general L_(p)-intersection bodies. In addition, associating with the L_(q)-radial combination and Lq-harmonic Blaschke combination, we establish the Brunn-Minkowski type inequalities of general Lp-intersection bodies for dual quermassintegrals, respectively. As applications, inequalities of volume are derived.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.
文摘Let M be a compact convex hypersurface of class C2, which is assumed to bound a nonempty convex body K in the Euclidean space Rn and H be the mean curvature of M. We obtain a lower bound of the total square of mean curvature fM H2dA The bound is the Minkowski quermassintegral of the convex body K. The total square of mean curvature attains the lower bound when M is an （n - 1）-sphere.
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.
基金Project supported in part by National Natural Science Foundation of China （Grant No. 10271071）
文摘The lower bound for the volume of the zonotope for John-basis had been given by Ball. In this paper, a simple proof of Ball＇s inequality was first provided, then the result of Ball was generalized from John-basis to a sequence of non-zero vectors which are full rank. Furthermore, the upper bound for the volumes of zonotopes was given. Finally the inequalities were deduced for the inradius and circumradius of a certain zonotope.
基金supported by National Natural Science Foundation of China(Grant No.11001163)Innovation Program of Shanghai Municipal Education Commission(Grant No.11YZ11)
文摘The notion of mixed quermassintegrals in the classical Brunn-Minkowski theory is extended to that of Orlicz mixed quermassintegrals in the Orlicz Brunn-Minkowski theory. The analogs of the classical Cauchy- Kuhota formula, the Minkowski isoperimetric inequality and the Brunn-Minkowski inequality are established for this new Orlicz mixed quermassintegrals.
基金This work was partially supported by the National Doctorial Discipline Development Foundation and Hunan Provincial Science Foundation.
文摘In this paper,we establish two theorems for the quermassintegrals of convex bodies,which are the generalizations of the well-known Aleksandrov's projection theorem and Loomis-Whitney's inequality,respectively.Applying these two theorems,we obtain a number of inequalities for the volumes of projections of convex bodies.Besides,we introduce the concept of the perturbation element of a convex body,and prove an extreme property of it.
基金supported by National Natural Science Foundation of China(Grant No.11471206)
文摘A class of geometric quantities for convex bodies is introduced iu the framework of Orlicz Brunn- Minkowski theory. It is shown that these new geometric quantities are affine invariant and precisely the generalizations of classical affine quermassintegrals.
基金This work was partially supported by the National Natural Science Foundation of China (Grant No.10271071)Zhejiang Provincial Natural Science Foundation of China (Grant No.Y605065)Foundation of the Education Department of Zhejiang Province of China (Grant No.20050392)
文摘In this paper,we first introduce a concept of L_p-dual Quermassintegral sum function of convex bodies and establish the polar projection Minkowski inequality and the polar projection Aleksandrov-Fenchel inequality for L_p-dual Quermassintegral sums.Moreover,by using Lutwak’s width-integral of index i,we establish the L_p-Brunn-Minkowski inequality for the polar mixed projec- tion bodies.As applications,we prove some interrelated results.
基金supported by National Natural Science Foundation of China (Grant No. 10971205)
文摘We establish the cyclic inequality for i-th L p-dual mixed volume and Lp-dual Urysohn inequality between p-mean width and Lp-dual quermassintegral. Moreover, the dual isoperimetric inequality for Lp-dual mixed volume is proved, which is an extension of the classical dual isoperimetric inequality.
基金Supported by the National Natural Science Foundation of China(11371224)Excellent Foundation of Graduate Student of China Three Gorges University(2017YPY077)
文摘Based on Lutwak＇s the notion of Lp-difference bodies, Wang and Ma introduced asymmetric Lp-difference bodies and gave their extremum values for volumes. In this paper, we establish the extremum value inequalities for the quermassintegrals and dual quermassintegrals of asymmetric Lp-difference bodies and their polars, respectively.
基金the National Natural Science Foundation of China(11371224)the Innovation Foundation of Graduate Student of China Three Gorges University(2019SSPY146)。
文摘In this paper, we study the extremum inequalities of general L_(p)-intersection bodies. In addition, associating with the L_(q)-radial combination and Lq-harmonic Blaschke combination, we establish the Brunn-Minkowski type inequalities of general Lp-intersection bodies for dual quermassintegrals, respectively. As applications, inequalities of volume are derived.
