Hermann minkowski 18641909 found that the theory of special relativity, introduced by his former student albert einstein, could be best understood as a fourdimensional space, since known as the minkowski spacetime. Inequality 11 is especially important when gx is a greens function. Liu, rotfeld inequality for partitioned matrices with numerical ranges in a sector, linear multilinear algebra. A special case of this general inequality, for which the equality conditions are known, is the minkowski inequality which states as follows. Many of them have been proved by famous mathematicians and named after them. A basis of m is called minkowski reduced if the following properties are satisfied.
On the other hand, liu 16 also gave an extension of ky fans inequality 3 and brunnminkowski type inequality 5 to the case of sector matrices. Bellmans inequality could actually give a simple proof of 5. In case 1 minkowskis inequality for integrals the following inequality is a generalization of minkowskis inequality c12. Some classical inequalities among all inequalities, there is a number of wellknown classical inequalities. The examples of twoqubit system and qudit with j 32 are discussed. S ze, determinantal and eigenvalue inequalities for matrices with numerical ranges in a. Proving that the pnorm is a norm is a little tricky and not particularly relevant to this course. Minkowski reduction of integral matrices article pdf available in mathematics of computation 33145 january 1979 with 98 reads how we measure reads. In the sections that follow, we prove theorem 2 without assuming minkowskis inequality. Minkowskitype inequality for arbitrary density matrices. Generalized minkowski inequality under appropriate conditions on the function h which appears below, and for 1 p 1. Yuan and leng 2007 gave a generalization of the matrix form of the brunnminkowski inequality. Cauchyschwarz inequality triangle inequality, reverse triangle inequality vector and matrix norms equivalence of p norms basic norm inequalities useful for proofs matrices 878o spring 2015 introduction to linear algebra january 26, 2017 1 22. Proof of minkowski s inequality this follows from holders inequality,and in my proof, for the sake of simplicity,ill use it minkowski s inequality states that.
In this note, we first give a simple proof of this inequality, and then show a generalization of this to a larger class of matrices, namely, matrices whose numerical ranges are contained in a sector. Minkowski in 1896 and expresses the fact that in ndimensional space, where the distance between the points x x 1, x 2. Notes on vector and matrix norms university of texas at austin. In this note, we first give an extension of yuanlengs result to multiple positive definite matrices, and then we further extend the result to a larger class of matrices whose numerical ranges are contained in a sector. Minkowski norms and holder inequality the family of minkowski norms 4 for a vector x e r is defined by imp li lxilp, 1 i p i 00, 8 il which, called a pnorm, includes all three norms discussed in the previous section as special cases, p 1,2 and 43. Abstract we develop a reverse entropy power inequality for convex measures, which may be. In each case equality holds if and only if the rows and are proportional. Notes on vector and matrix norms university of texas at. Feb 28, 2020 in this note, we first give an extension of yuanlengs result to multiple positive definite matrices, and then we further extend the result to a larger class of matrices whose numerical ranges are contained in a sector.
This note has been taken almost verbatim from existing notes of alex iosevich. Inequality 4 reverses the sign if 0 inequality 11 reverses the sign if 0 minkowski. Pdf a simple proof of the holder and the minkowski inequality. Extensions of brunnminkovskis inequality to multiple matrices. Pdf extensions of brunnminkovskis inequality to multiple. A generalization of the matrix form of the brunnminkowski.
Minkowskis inequality for integrals the following inequality is a generalization of minkowskis inequality c12. Generalizations of the brunnminkowski inequality sciencedirect. I would like to know whether anyone knows of any reverse forms of these inequalities. More generalizations of hartfiels inequality and the. Minkowski s inequality can be generalized in various ways also called minkowski inequalities.
Mar 04, 2005 in this paper, we establish an extension of the matrix form of the brunnminkowski inequality. We only need to prove the ag inequality because the hg inequality follows from the ag inequality and properties of the means ha 1 a 1 a. In some sense it is also a theorem on the change of the order of iterated integrals, but equality is only obtained if p1. A determinantal inequality for positive semidefinite matrices. By using minkowski inequality 37, we find an upper bound for the cost function q x. See minkowskis first inequality for convex bodies for minkowskis inequality in convex geometry. Does there exist a generalized version of the minkowski determinant inequality such that matrices a and b above could be any symmetric matrices without the need of positive definiteness. For minkowski s inequality is called the triangle inequality. For minkowskis inequality is called the triangle inequality.
The minkowski inequality has analogs for infinite series and integrals. A brunnminkowski inequality for the hessian eigenvalue in. In mathematical analysis, the minkowski inequality establishes that the l p spaces are normed vector spaces. Instead, we prove both the existence theorem ofminkowskiand his mixed volume inequality simultaneously, thereby. Clearly, the 1norm and 2 norms are special cases of the pnorm. Lecture notes topics in theoretical computer science. In mathematical physics, minkowski space or minkowski spacetime is a combination of threedimensional euclidean space and. On minkowskis inequality and its application journal of. As applications, we give generalizations on the metric addition inequality of alexander. By a standard continuity argument, we may assume c.
Let b be a positive semidefinite nsquare hermitian matrix o f rank r. In the last section, we prove the brunnminkowski inequality and characterize the equality case. More generalizations of hartfiels inequality and the brunn. Matrix inequalities in the theory of mixed quermassintegrals. A simple proof of the holder and the minkowski inequality. Some problems about matrices university of warwick.
The two fundamental inequalities are the minkowski and brunnminkowski inequalities. Minkowski inequality article about minkowski inequality by. In the paper, we first give an improvement of minkowski integral inequality. Instead, we prove both the existence theorem ofminkowskiand his mixed volume inequality simultaneously, thereby demonstrating the fundamental. The cauchyschwarz inequality and some simple consequences neil lyall abstract. Concentration inequalities higher school of economics.
Our result improves a recent result of liu linear algebra appl. On the other hand, liu 16 also gave an extension of ky fans inequality 3 and brunn minkowski type inequality 5 to the case of sector matrices. Thus, we need to show compute a determinantal inequality for positive definite matrices 825 proof of theorem 1. The two fundamental inequalities are the minkowski and brunn minkowski inequalities. Reverse brunnminkowski and reverse entropy power inequalities for convex measures sergey bobkova, mokshay madimanb, aschool of mathematics, university of minnesota, 206 church st. Hermann minkowski was born in the town of aleksota, the suwalki governorate, the kingdom of poland, part of the russian empire, lithuania, to lewin boruch minkowski, a merchant who subsidized the building of the choral synagogue in kovno, and rachel taubmann, both of jewish descent. Generalized minkowski inequality under appropriate conditions on the function h which appears below, and for 1 p minkowski inequality other similar results may be obtained by applying theorem 1 to suitable positive semidefinite hermitian commuting matrices a and b. To prove the triangle inequality requires the following classical result. For bipartite systems, this inequality coincides with a known entropic inequality like the subadditivity condition. As an application, we get new brunnminkowskitype inequalities for dual mixed volumes.
Minkowski reduction of integral matrices 203 definition. A constant rank theorem in this section we establish a constant rank theorem for the convex solution of the related nonlinear elliptic equation. Minkowskis inequality can be generalized in various ways also called minkowski inequalities. We discuss cardinality estimates, the structure of a random matrix from the set, discrete versions of the brunn minkowski inequality and the statistical dependence between row and column sums. In this note, we give some other generalizations of hartfiels inequality and the brunnminkowski inequality to sector matrices, the results obtained improve those of lin arch math 104. The aim of this note is to give another proof of these classical inequalities. We discuss cardinality estimates, the structure of a random matrix from the set, discrete versions of the brunnminkowski inequality and the statistical dependence between row and column sums. We obtain a new matrix inequality for an arbitrary density matrix of compositenoncomposite qudit systems including a singlequdit state. Proof of minkowskis inequality this follows from holders inequality,and in my proof,for the sake of simplicity,ill use it. Extensionsofbrunnminkovskisinequalitytomultiple matrices. As an application, we get new brunn minkowski type inequalities for dual mixed volumes.
The cases p 1 and p 1 follow from the triangle inequality. Theorem 4 extends the classical minkowski determinant inequality to all of the coefficients in the characteristic polynomial and theorem 5 extends a recent concavity result of fan 1 for determinants. The cauchyschwarz inequality let x and y be points in the euclidean space rn which we endow with the usual inner product and norm, namely x,y xn j1 x jy j and kxk xn j1 x2 j. This page is about minkowskis inequality for norms. These include, in particular, bernoullis, youngs, holders, cauchyschwarz, and minkowskis inequalities of course, this is not a complete list. Im not sure whether the second has a name, but it can be found, e.
613 6 84 658 804 1088 721 474 553 1284 444 1073 745 1486 377 1279 1147 673 144 656 260 1588 1223 1220 564 749 952 540 285 1512 1118 1359 16 662 1333 472 1324 1354 1316 1372 455 463 348 1145 497 474 958 67