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.
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
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
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 article, we study isoperimetric problems of the calculus of variations with left and right Riemann-Liouville fractional derivatives. Both situations when the lower bound of the variational integrals coincide a...In this article, we study isoperimetric problems of the calculus of variations with left and right Riemann-Liouville fractional derivatives. Both situations when the lower bound of the variational integrals coincide and do not coincide with the lower bound of the fractional derivatives are considered.展开更多
In this paper,we give some isoperimetric upper bounds for the first eigenvalue of the p-biharmonic operator of an n-dimensional embedded closed hypersurface in an Euclidean space.We also give Reilly-type inequalities ...In this paper,we give some isoperimetric upper bounds for the first eigenvalue of the p-biharmonic operator of an n-dimensional embedded closed hypersurface in an Euclidean space.We also give Reilly-type inequalities for the first eigenvalue of the p-biharmonic operator of an n-dimensional closed submanifold immersed into a higher dimensional manifold such as an Euclidean space,a unit sphere,a projective space.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, homotopy perturbation method is applied to solve moving boundary and isoperimetric problems. This method does not depend upon a small parameter in the equation. homotopy is constructed with an imbedding...In this paper, homotopy perturbation method is applied to solve moving boundary and isoperimetric problems. This method does not depend upon a small parameter in the equation. homotopy is constructed with an imbedding parameter p, which is considered as a “small parameter”. Finally, we use combined homotopy perturbation method and Green’s function method for solving second order problems. Some examples are given to illustrate the effectiveness of methods. The results show that these methods provides a powerful mathematical tools for solving problems.展开更多
This article deals with the sharp discrete isoperimetric inequalities in analysis and geometry for planar convex polygons.First,the analytic isoperimetric inequalities based on the Schur convex function are establishe...This article deals with the sharp discrete isoperimetric inequalities in analysis and geometry for planar convex polygons.First,the analytic isoperimetric inequalities based on the Schur convex function are established.In the wake of the analytic isoperimetric inequalities,Bonnesen-style isoperimetric inequalities and inverse Bonnesen-style inequalities for the planar convex polygons are obtained.展开更多
In this paper,we prove the Wulff–Gage isoperimetric inequality for origin-symmetric convex bodies and the uniqueness of the log-Minkowski problem in R2.Then we give a new proof of the log-Minkowski inequality of curv...In this paper,we prove the Wulff–Gage isoperimetric inequality for origin-symmetric convex bodies and the uniqueness of the log-Minkowski problem in R2.Then we give a new proof of the log-Minkowski inequality of curvature entropy for origin-symmetric convex bodies with C2boundaries.展开更多
In this paper,we give a reverse analog of the Bonnesen-style inequality of a convex domain in the surface X of constant curvature,that is,an isoperimetric deficit upper bound of the convex domain in X.The result is an...In this paper,we give a reverse analog of the Bonnesen-style inequality of a convex domain in the surface X of constant curvature,that is,an isoperimetric deficit upper bound of the convex domain in X.The result is an analogue of the known Bottema's result of 1933 in the Euclidean plane E2.展开更多
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
This paper deals with the following isoperimetric problem in the plane:Among all regions with prescribed perimeter and covering a given line segment,what is the region that has the greatest area?
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
On surfaces we give conditions under which the solution of a restricted local isoperimetric problem for sectors with small solid angle is the circular sector and we characterize these surfaces. Also we study this prob...On surfaces we give conditions under which the solution of a restricted local isoperimetric problem for sectors with small solid angle is the circular sector and we characterize these surfaces. Also we study this problem for general spherical cones on hypersurfaces in higher dimensional Riemannian manifolds.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
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
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.
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 by FEDER funds through COMPETE - Operational Programme Factors of Competitiveness("Programa Operacional Factores de Competitividade")Portuguese funds through the Center for Research and Development in Mathematics and Applications(University of Aveiro) and the Portuguese Foundation for Science and Technology("FCT - Fundao para a Ciencia e a Tecnologia"),within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690
文摘In this article, we study isoperimetric problems of the calculus of variations with left and right Riemann-Liouville fractional derivatives. Both situations when the lower bound of the variational integrals coincide and do not coincide with the lower bound of the fractional derivatives are considered.
基金Supported by the National Natural Science Foundation of China(Grant No.12071051).
文摘In this paper,we give some isoperimetric upper bounds for the first eigenvalue of the p-biharmonic operator of an n-dimensional embedded closed hypersurface in an Euclidean space.We also give Reilly-type inequalities for the first eigenvalue of the p-biharmonic operator of an n-dimensional closed submanifold immersed into a higher dimensional manifold such as an Euclidean space,a unit sphere,a projective space.
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, homotopy perturbation method is applied to solve moving boundary and isoperimetric problems. This method does not depend upon a small parameter in the equation. homotopy is constructed with an imbedding parameter p, which is considered as a “small parameter”. Finally, we use combined homotopy perturbation method and Green’s function method for solving second order problems. Some examples are given to illustrate the effectiveness of methods. The results show that these methods provides a powerful mathematical tools for solving problems.
基金Supported by NSFC(Grant No.12141101)Natural Science Foundation Project of Chongqing(Grant No.CSTB2024NSCQ-MSX0937)Technology Research Foundation of Chongqing Educational committee(Grant No.KJZD-K202200509)。
文摘This article deals with the sharp discrete isoperimetric inequalities in analysis and geometry for planar convex polygons.First,the analytic isoperimetric inequalities based on the Schur convex function are established.In the wake of the analytic isoperimetric inequalities,Bonnesen-style isoperimetric inequalities and inverse Bonnesen-style inequalities for the planar convex polygons are obtained.
基金supported by the Characteristic Innovation Projects of Universities in Guangdong Province(Grant No.2023KTSCX381)supported by the Major Special Project of NSFC(Grant No.12141101)+2 种基金the Science and Technology Research Foundation of Chongqing Educational Committee(Grant Nos.KJZD-K202200509KJZD-K202500503)the Natural Science Foundation Project of Chongqing(Grant No.CSTB2024NSCQMSX0937)。
文摘In this paper,we prove the Wulff–Gage isoperimetric inequality for origin-symmetric convex bodies and the uniqueness of the log-Minkowski problem in R2.Then we give a new proof of the log-Minkowski inequality of curvature entropy for origin-symmetric convex bodies with C2boundaries.
基金supported in part by National Natural Science Foundation of China (Grant No.10971167)
文摘In this paper,we give a reverse analog of the Bonnesen-style inequality of a convex domain in the surface X of constant curvature,that is,an isoperimetric deficit upper bound of the convex domain in X.The result is an analogue of the known Bottema's result of 1933 in the Euclidean plane E2.
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the National Natural Science Foundation of China(Grant No.10671066)the Shanghai Leading Academic Discipline Project(Project No.B407)partially supported by the National Key Basic Research Project of China(Grant No.2006CB805902)
文摘This paper deals with the following isoperimetric problem in the plane:Among all regions with prescribed perimeter and covering a given line segment,what is the region that has the greatest area?
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.
基金supported by a DGI(Spain) and Feder Project MTM 2004-06015-C02-01 AVCiT GRUPOS 03/169
文摘On surfaces we give conditions under which the solution of a restricted local isoperimetric problem for sectors with small solid angle is the circular sector and we characterize these surfaces. Also we study this problem for general spherical cones on hypersurfaces in higher dimensional Riemannian manifolds.
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.
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.
