In this paper,by the theory of geometric inequalities,some new Bonnesenstyle isoperimetric inequalities of n-dimensional simplex are proved.In several cases,these inequalities imply characterizations of regular simplex.
In this paper,we investigate the translative containment measure for a convex domain K_i to contain,or to be contained in the homothetic copy of another convex domain tK_j(t≥0).Via the formulas of translative Blaschk...In this paper,we investigate the translative containment measure for a convex domain K_i to contain,or to be contained in the homothetic copy of another convex domain tK_j(t≥0).Via the formulas of translative Blaschke and Poincare in integral formula,we obtain a Bonnesen-style symmetric mixed isohomothetic inequality.The Bonnesen-style symmetric mixed isohomothetic inequality obtained is known as Bonnesen-style inequality if one of the domains is a disc.As a direct consequence,we attain an inequality which strengthen the result proved by Bonnesen,Blaschke and Flanders.Furthermore,by the containment measure and Blaschke’s rolling theorem,we obtain the reverse Bonnesen-style symmetric mixed isohomothetic inequalities.These inequalities are the analogues of the known Bottema’s result in 1933.展开更多
In this article, we prove certain isoperimetric inequalities for eigenvalues of Riesz potentials and show some applications of the results to a non-local boundary value problem of the Laplace operator.
In this paper,we establish a stability estimate for the isoperimetric inequality of horospherically convex domains in hyperbolic plane.This estimate involves a relationship between the Hausdorff distance to a geodesic...In this paper,we establish a stability estimate for the isoperimetric inequality of horospherically convex domains in hyperbolic plane.This estimate involves a relationship between the Hausdorff distance to a geodesic ball and the deficit in the isoperimetric inequality,where the coefficient of the deficit is a universal constant.展开更多
Isoperimetric type inequalities for integral geometric invariants of random lines in the Euclidean space are shown.Entropy inequalities of probability densities on the affine Grassmann manifold of lines are given.
We first estimate the containment measure of a convex domain to contain in another in a surface X of constant curvature.Then we obtain the analogue of the known Bonnesen isoperimetric inequality for convex domain in X...We first estimate the containment measure of a convex domain to contain in another in a surface X of constant curvature.Then we obtain the analogue of the known Bonnesen isoperimetric inequality for convex domain in X.Finally we strengthen the known Bonnesen isoperimetric inequality.展开更多
Lutwak, Yang, and Zhang posed the notion of Lp-curvature images and established several Lp analogs of the affne isoperimetric inequality. In this article, the notion of Lp-mixed curvature images is introduced, Lp-curv...Lutwak, Yang, and Zhang posed the notion of Lp-curvature images and established several Lp analogs of the affne isoperimetric inequality. In this article, the notion of Lp-mixed curvature images is introduced, Lp-curvature images being a special case. The properties and Lp analogs of the affne isoperimetric inequality are established for Lp-mixed curvature images.展开更多
A lot of forms and applications have been found in Poincar6 inequality, but the optimum constants satisfying Poincar6 inequality have not been estimated. This paper estimates the optimum constants λ0 and λ1 satisfyi...A lot of forms and applications have been found in Poincar6 inequality, but the optimum constants satisfying Poincar6 inequality have not been estimated. This paper estimates the optimum constants λ0 and λ1 satisfying Poincaré inequality by using isoperimetric constants. It is λ0≥k0^2/（2R） and λ1 ≥k1^2/（2R）.展开更多
The author proves that the isoperimetric inequality on the graphic curves over circle or hyperplanes over S^(n-1) is satisfied in the cigar steady soliton and in the Bryant steady soliton.Since both of them are Rieman...The author proves that the isoperimetric inequality on the graphic curves over circle or hyperplanes over S^(n-1) is satisfied in the cigar steady soliton and in the Bryant steady soliton.Since both of them are Riemannian manifolds with warped product metric,the author utilize the result of Guan-Li-Wang to get his conclusion.For the sake of the soliton structure,the author believes that the geometric restrictions for manifolds in which the isoperimetric inequality holds are naturally satisfied for steady Ricci solitons.展开更多
This is a survey of the results in[14]regarding the isoperimetric problem in the Riemannian manifold.We consider a mean curvature type flow in the Riemannian manifold endowed with a non-trivial conformal vector field,...This is a survey of the results in[14]regarding the isoperimetric problem in the Riemannian manifold.We consider a mean curvature type flow in the Riemannian manifold endowed with a non-trivial conformal vector field,which was firstly introduced by Guan and Li[8]in space forms.This flow preserves the volume of the bounded domain enclosed by a star-shaped hypersurface and decreases the area of hypersurface under certain conditions.We will prove the long time existence and convergence of the flow.As a result,the isoperimetric inequality for such a domain is established.展开更多
We discuss the higher dimensional Bonnesen-style inequalities. Though there are many Bonnesen-style inequalities for domains in the Euclidean plane R2 few results for general domain in Rn （n ≥ 3） are known. The res...We discuss the higher dimensional Bonnesen-style inequalities. Though there are many Bonnesen-style inequalities for domains in the Euclidean plane R2 few results for general domain in Rn （n ≥ 3） are known. The results obtained in this paper are for general domains, convex or non-convex, in Rn.展开更多
This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First,the author gives a new and simple proof of a lower bou...This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First,the author gives a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric inequality in convex cones. For this last inequality, the new proof was recently found by the author, Xavier Ros-Oton, and Joaquim Serra in a work where new Sobolev inequalities with weights came up by studying an open question raised by Haim Brezis.展开更多
For a positive integer s,the projection body of an s-concave function f:R^(n)→[0,+∞),a convex body in the(n+s)-dimensional Euclidean space R^(n+s),is introduced.Associated inequalities for s-concave functions,such a...For a positive integer s,the projection body of an s-concave function f:R^(n)→[0,+∞),a convex body in the(n+s)-dimensional Euclidean space R^(n+s),is introduced.Associated inequalities for s-concave functions,such as,the functional isoperimetric inequality,the functional Petty projection inequality and the functional Loomis-Whitney inequality are obtained.展开更多
基金The Doctoral Programs Foundation(20113401110009)of Education Ministry of ChinaUniversities Natural Science Foundation(KJ2016A310)of Anhui Province
文摘In this paper,by the theory of geometric inequalities,some new Bonnesenstyle isoperimetric inequalities of n-dimensional simplex are proved.In several cases,these inequalities imply characterizations of regular simplex.
基金supported in part by the National Natural Science Foundation of China(11801048)the Natural Science Foundation Project of CSTC(cstc2017jcyjAX0022)Innovation Support Program for Chongqing overseas Returnees(cx2018034)
文摘In this paper,we investigate the translative containment measure for a convex domain K_i to contain,or to be contained in the homothetic copy of another convex domain tK_j(t≥0).Via the formulas of translative Blaschke and Poincare in integral formula,we obtain a Bonnesen-style symmetric mixed isohomothetic inequality.The Bonnesen-style symmetric mixed isohomothetic inequality obtained is known as Bonnesen-style inequality if one of the domains is a disc.As a direct consequence,we attain an inequality which strengthen the result proved by Bonnesen,Blaschke and Flanders.Furthermore,by the containment measure and Blaschke’s rolling theorem,we obtain the reverse Bonnesen-style symmetric mixed isohomothetic inequalities.These inequalities are the analogues of the known Bottema’s result in 1933.
文摘In this article, we prove certain isoperimetric inequalities for eigenvalues of Riesz potentials and show some applications of the results to a non-local boundary value problem of the Laplace operator.
文摘In this paper,we establish a stability estimate for the isoperimetric inequality of horospherically convex domains in hyperbolic plane.This estimate involves a relationship between the Hausdorff distance to a geodesic ball and the deficit in the isoperimetric inequality,where the coefficient of the deficit is a universal constant.
文摘Isoperimetric type inequalities for integral geometric invariants of random lines in the Euclidean space are shown.Entropy inequalities of probability densities on the affine Grassmann manifold of lines are given.
基金supported by National Natural Science Foundation of China (Grant No. 10971167)
文摘We first estimate the containment measure of a convex domain to contain in another in a surface X of constant curvature.Then we obtain the analogue of the known Bonnesen isoperimetric inequality for convex domain in X.Finally we strengthen the known Bonnesen isoperimetric inequality.
基金Supported by Innovation Program of Shanghai Municipal Education Commission (10YZ160)Science and Technology Commission Foundation of Shanghai (071605123)Science Foundation for the Excellent Youth Scholars of Shanghai
文摘Lutwak, Yang, and Zhang posed the notion of Lp-curvature images and established several Lp analogs of the affne isoperimetric inequality. In this article, the notion of Lp-mixed curvature images is introduced, Lp-curvature images being a special case. The properties and Lp analogs of the affne isoperimetric inequality are established for Lp-mixed curvature images.
基金The Science and Technology Foundation of Chongqing Municipal Education Commission (No.KJ071106)
文摘A lot of forms and applications have been found in Poincar6 inequality, but the optimum constants satisfying Poincar6 inequality have not been estimated. This paper estimates the optimum constants λ0 and λ1 satisfying Poincaré inequality by using isoperimetric constants. It is λ0≥k0^2/（2R） and λ1 ≥k1^2/（2R）.
基金This work was supported by the National Natural Science Foundation of China(Nos.11721101,11526212).
文摘The author proves that the isoperimetric inequality on the graphic curves over circle or hyperplanes over S^(n-1) is satisfied in the cigar steady soliton and in the Bryant steady soliton.Since both of them are Riemannian manifolds with warped product metric,the author utilize the result of Guan-Li-Wang to get his conclusion.For the sake of the soliton structure,the author believes that the geometric restrictions for manifolds in which the isoperimetric inequality holds are naturally satisfied for steady Ricci solitons.
文摘This is a survey of the results in[14]regarding the isoperimetric problem in the Riemannian manifold.We consider a mean curvature type flow in the Riemannian manifold endowed with a non-trivial conformal vector field,which was firstly introduced by Guan and Li[8]in space forms.This flow preserves the volume of the bounded domain enclosed by a star-shaped hypersurface and decreases the area of hypersurface under certain conditions.We will prove the long time existence and convergence of the flow.As a result,the isoperimetric inequality for such a domain is established.
基金supported by National Natural Science Foundation of China (Grant No. 10971167)
文摘We discuss the higher dimensional Bonnesen-style inequalities. Though there are many Bonnesen-style inequalities for domains in the Euclidean plane R2 few results for general domain in Rn （n ≥ 3） are known. The results obtained in this paper are for general domains, convex or non-convex, in Rn.
文摘This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First,the author gives a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric inequality in convex cones. For this last inequality, the new proof was recently found by the author, Xavier Ros-Oton, and Joaquim Serra in a work where new Sobolev inequalities with weights came up by studying an open question raised by Haim Brezis.
基金supported by the National Natural Science Foundation of China(Nos.12001291,12071318)Chern Institute of Mathematics,Nankai University。
文摘For a positive integer s,the projection body of an s-concave function f:R^(n)→[0,+∞),a convex body in the(n+s)-dimensional Euclidean space R^(n+s),is introduced.Associated inequalities for s-concave functions,such as,the functional isoperimetric inequality,the functional Petty projection inequality and the functional Loomis-Whitney inequality are obtained.
