Informationtheoretical inequalities for logconcave and stable densities giuseppe toscani department of mathematics university of pavia, italy nonlocal nonlinear partial di erential equations and applications anacapri, september 1418, 2015. Informationtheoretical inequalities for logconcave and. In particular, in all examples considered, we find that for fixed volume of the observable universe, the entropy is increased by adding black holes. Isoperimetric problems are a trademark topic in mathematical physics. Domoshnitsky, ariel university center of samaria, israel jewgeni.
The equality holds only when is a ball in on a plane, i. Huisken has observed that arnowitt, deser and misner adm mass of an asymptotically flat af manifold see definition 1. General relativity, differential geometry in physics. Isoperimetric inequalities and the concept of mass in general relativity abstract. Brascampliebluttinger inequalities for convex domains of finite inradius mendezhernandez, pedro j. Thirteen basic inequalities relating tail area probabilities to moments are stated. We shall consider first a closed surface a surrounding a. Kinetic equations, manybody systems and functional inequalities. Requiring only of a basic course in differential geometry, this. Am27, volume 27 annals of mathematics studies 27 on free shipping on qualified orders. Their work has attracted a number of mathematicians into this area of study and as connections among isoperimetric inequalities, a. In this paper, weshow that the socalled sneakout principle for discrete inequalities is valid also on a general time scale.
Mathematical physics books, journals, series, textbooks, reference works, electronic media from springer. Journal of inequalities and applications editorial board. American mathematical monthlythe purpose of this text is to lay a broad foundation for an understanding of the problems of the calculus of. Isoperimetric inequality on asymptotically flat manifolds. The book description for the forthcoming isoperimetric inequalities in mathematical physics.
Am27, volume 27 annals of mathematics studies 27 paperback august 21, 1951 by g. Yucheng an, hairong liu, long tian the dirichlet problem for a subelliptic equation with singular nonlinearity on the heisenberg group. Inequalities for means of chords, with application to. Domoshnitsky, ariel university center of samaria, israel jewgeni dshalalow, florida institute. Harrell c, michael loss a department of theoretical physics, nuclear physics institute, academy of. Inequalities and uncertainty principles journal of mathematical physics 19, 461 1978. Shusen ding, seattle university, united states of america editorial board. Journal of mathematical inequalities rg journal impact. Thermodynamic volumes and isoperimetric inequalities. A rigorous proof of the classical isoperimetric inequality for was given by f. The isoperimetric inequality and isoperimetric surfaces have a very long history and many important applications in mathematics, see, e. The books strategy works for differential and integral equations and systems and for many theoretical and applied problems in mathematics, mathematical physics. Isoperimetric inequalities in mathematics, the isoperimetric inequality is a geometric inequality involving the square of the circumference of a closed curve in the plane and the area of a plane region it encloses, as well as its various generalizations.
Isoperimetric and other inequalities in the theory of neutron. The journal of mathematical inequalities jmi presents carefully selected original research articles from all areas of pure and applied mathematics, provided they are concerned with mathematical inequalities and their numerous applications. Am27, volume 27 by georg polya, 9780691079882, available at book depository with free delivery worldwide. In graph theory, isoperimetric inequalities are at the heart of the study of expander graphs, which are sparse graphs that have strong connectivity properties. The rst and best known isoperimetric inequality is the classical isoperimetric inequality a l2 4 relating the area aenclosed by a planar closed curve of perimeter l. Diazmetcalf and polyaszego type inequalities associated with saigo fractional integral operator saxena, ram k. By means of an iteration process and symmetrization methods the solution in an arbitrary domain is compared with the one for the sphere of the same volume.
The calculus of variations evolved from attempts to solve this problem and the. In the absence of any restriction on shape, the curve is a circle. The isoperimetric inequality 1 is valid also for a twodimensional manifold of bounded curvature, which is a more general type of manifold than a riemannian manifold. Aug 11, 2008 these inequalities are derived from commutator inequalities related to the heisenberg uncertainty principle. Inequalities for means of chords, with application to isoperimetric problems pavel exnera,b, evans m. The stong isoperimetric inequality of bonnesen andrejs treibergs university of utah among all simple closed curves in the plane with a. Isoperimetric inequalities in mathema tical physics. Related books essential discrete mathematics for computer science harry lewis and rachel zax. American mathematical monthlythe purpose of this text is to lay a broad foundation for an understanding of the problems of the calculus of variations. In this paper we introduce the concept of area functions for a given function.
Isoperimetric inequalities for the logarithmic potential. We will also show that, in a way, steiner symmetrization could be used as a useful tool to prove pettys conjectured projection inequality. Expander constructions have spawned research in pure and applied mathematics, with several applications to complexity theory, design of robust computer networks, and the theory of errorcorrecting codes. Isoperimetric inequality, classical encyclopedia of. Harrell c, michael loss a department of theoretical physics, nuclear physics institute, academy of sciences, 25068 re. The inequalities proved provide upper and lower bounds for the. The quantities involved in these inequalities all refer to bare solids with isotropic scattering and are. Buy isoperimetric inequalities in mathematical physics.
Existence theorems and a priori bounds for a class of nonlinear parabolic equations are established. They also paved the way to the understanding of many related variational problems in mathematical. Some isoperimetric inequalities in the torsion problem for multiply connected regions, studies in mathematical analysis and related topics. Isoperimetric inequalities for eigenvalues of the laplacian. Journal of mathematical analysis and applications 352. Annals of mathematics studies 27 book 27 paperback. Isoperimetric and other inequalities in the theory of. Szeg o author see all 2 formats and editions hide other formats and editions. Communications in mathematical physics editors springer. Lecture 21 inequalities another way in which optimization can be applied is to prove inequalities. Isoperimetric inequalities in mathematical physics princeton u. Using 1, inequalities can be established for the length of a. The calculus of variations evolved from attempts to solve this problem and the brachistochrone leasttime problem in 1638 the italian mathematician and astronomer. Percolation on finite graphs and isoperimetric inequalities.
Cms treatises in mathematics canadian mathematical society. Princeton university press august 21, 1951 language. Isoperimetric inequalities for a class of nonlinear. We will present a survey on isoperimetric inequalities in real, finitedimensional banach spaces, also called minkowski spaces. Studies in mathematical physics princeton university press. Ladislav matejicka proof of one open inequality of laubilani type. For the logarithmic potential operator on bounded open or triangular domains, we also obtain analogies of the rayleighfaberkrahn or polya inequalities, respectively. Some isoperimetric and other inequalities occurring in the one.
Cms treatises in mathematics is a new series of books. In the order of treatment, figures in space will precede figures in a plane. Isoperimetric inequalities for schatten norms of riesz. Isoperimetric inequalities and their applications siam. These treatises emphasize the interdisciplinary character of the mathematical sciences and facilitate integration of methods and. The application of isoperimetric inequalities for nonlinear. The classical isoperimetric inequality gives a solution of the isoperimetric problem. Applications of such inequalities can be found in stochastic geometry, functional analysis, fourier analysis, mathematical physics, discrete geometry, integral geometry, and various further mathematical disciplines. There are many results that focus on large isoperimetric regions in. Isoperimetric inequalities in mathematical physics annals. The purpose of this expository paper is to collect some mainly recent inequalities, conjectures, and open questions closely related to isoperimetric problems in real, finitedimensional banach spaces minkowski spaces. Equality in 1 is attained for a nonregular object a domain isometric to the lateral surface of a right circular cone with complete angle about the vertex. Purchase mathematical inequalities, volume 67 1st edition.
Isoperimetric inequalities in mathematical physics book. Isoperimetric inequalities in mathematical physics szego g. In their famous book isoperimetric inequalities in mathematical physics, polya and szego extended this notion to include inequalities for domain functionals, provided that the equality sign is attained for some domain or in the limit as the domain degenerates 15. In dimensional space the inequality lower bounds the surface area or perimeter of a set. Victor garcia garcia, pedro ortega salvador weighted inequalities for the multilinear hilbert and calderon operators and applications. Isoperimetric inequalities in mathematical physics annals of. Isoperimetric problem, in mathematics, the determination of the shape of the closed plane curve having a given length and enclosing the maximum area. Ivan area, university of vigo, spain mircea balaj, university of oradea, romania martin bohner, missouri university of science and technology, united states of america.
Isoperimetric definition of isoperimetric by the free. Their work has attracted a number of mathematicians into this area of study and as connections among isoperimetric inequalities, a priori bounds, and. These treatises emphasize the interdisciplinary character of the mathematical sciences and facilitate integration of methods and results from different areas of current research. A family of optimal conditions for the absence of bound states in a potential, in studies in mathematical physics, edited by e. Buy isoperimetric inequalities in mathematical physics annals of mathematics studies by george polya, gabor szego isbn. Isoperimetric inequalities have a long history in geometry and analysis, motivating conceptual progress in these areas of mathematics for many centuries. For the classical isoperimetric inequality was known in antiquity. Isoperimetric inequalities in mathematical physics szego. Some new h p,l p type inequalities of maximal operators of vilenkinnorlund means with nondecreasing coefficients 10551069 view. In this paper we prove that the disc is a maximiser of the schatten pnorm of the logarithmic potential operator among all domains of a given measure in r 2, for all even integers 2. It is shown that among all domains of given volume the sphere is the least stable. Probability inequalities of the tchebycheff type nist page. In this note we prove that the ball is a maximiser for integer order schatten pnorms of the riesz potential operators among all domains of a given measure in r d.
The isoperimetric inequality 1 is valid also for a twodimensional manifold of bounded curvature, which is a more general type of manifold than a riemannian manifold equality in 1 is attained for a nonregular object a domain isometric to the lateral surface of a right circular cone with complete angle about the ver. Everyday low prices and free delivery on eligible orders. Isoperimetric inequalities have played an important role in mathematics since the times of ancient greece. Journal of mathematical inequalities volume 10, number 2 2016, 393403 doi. The logarithmic potential operator can be related to a nonlocal boundary value problem for the laplacian, so we obtain isoperimetric inequalities for its eigenvalues as well. Journal of research of the national bureau of standardsb. The author discusses inequalities in euclidean and riemannian geometry, methods of classical differential geometry and elementary modern geometric measure, discretization of smooth spaces, and the influence of isoperimetric inequalities on heat diffusion on riemannian manifolds. Mia, oam, jmi, dea, fdc, jca and their logos are trademarks owned by the element. We shall examine geometrical and physical quantities functionals depending on the shape and size of a closed surface, or of a closed curve.
Isoperimetric synonyms, isoperimetric pronunciation, isoperimetric translation, english dictionary definition of isoperimetric. The isoperimetric inequality, the mathematical statememt of this fact, is done in two parts. In particular, the result is valid for the polyharmonic newton potential operator, which is related to a nonlocal boundary value problem for the polylaplacian extending the one considered by m. In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. Isoperimetric inequalities in mathematical physics. Ivan area, university of vigo, spain mircea balaj, university of oradea, romania martin bohner, missouri university of science and technology, united states of america daniel breaz, university of alba iulia, romania alexander i. My motivation comes from a problem in mathematical physics where an isoperimetric inequality is deduced so i dont want to assume the isoperimetric inequality beforehand. Daniel breaz, university of alba iulia, romania alexander i. We then apply these area functions to establish new analytic and geometric isoperimetric inequalities. Dec 22, 2004 some isoperimetric and other inequalities occurring in the one.
1308 106 947 25 323 260 435 1447 1003 1557 35 857 1351 1070 767 1130 1421 1428 572 1241 983 989 585 925 88 1435 1120 4 198 1257 1165