some results on almost square banach spaces abstract we study almost square banach spaces under a topological point of view indeed we prove that the class of banach spaces which admits an equivalent norm to be asq is that of those banach spaces which contain an isomorphic copy of c 0 we also prove that the symmetric projective tensor products of an almost square banach space have the strong diameter two property

some remarks on quasiconvexity and rank one convexity we explore some necessary conditions for quasiconvexity in an attempt to show that rank one convexity does not imply quasiconvexity when the target space for deformations is two dimensional an interesting construction is presented showing how rank one directions may fit with each other making the task harder than in higher dimensions

characterizations of symmetric polyconvexity symmetric quasiconvexity plays a key role for energy minimization in geometrically linear elasticity theory due to the complexity of this notion a common approach is to retreat to necessary and sufficient conditions that are easier to handle this article focuses on symmetric polyconvexity which is a sufficient condition we prove a new characterization of symmetric polyconvex functions in the two and three dimensional setting and use it to investigate relevant subclasses like symmetric polyaffine functions and symmetric polyconvex quadratic forms in particular we provide an example of a symmetric rank one convex quadratic form in 3d that is not symmetric polyconvex the construction takes the famous work by serre from 1983 on the classical situation without symmetry as inspiration beyond their theoretical interest these findings may turn out useful for computational relaxation and homogenization

an isometric study of the lindeberg feller central limit theorem via stein s method a b s t r a c t we use stein s method to prove a generalization of the lindeberg feller clt providing an upper and a lower bound for the superior limit of the kolmogorov distance between a normally distributed random variable and the rowwise sums of a rowwise independent triangular array of random variables which is asymptotically negligible in the sense of feller a natural example shows that the upper bound is of optimal order the lower bound improves a result by andrew barbour and peter hall

almost square dual banach spaces abstract we show that finite dimensional banach spaces fail to be uniformly non locally almost square moreover we construct an equivalent almost square bidual norm on l as a consequence we get that every dual banach space containing c 0 has an equivalent almost square dual norm finally we characterize separable real almost square spaces in terms of their position in their fourth duals

octahedral norms and convex combination of slices in banach spaces abstract we study the relation between octahedral norms daugavet property and the size of convex combinations of slices in banach spaces we prove that the norm of an arbitrary banach space is octahedral if and only if every convex combination of w slices in the dual unit ball has diameter 2 which answers an open question as a consequence we get that the banach spaces with the daugavet property and its dual spaces have octahedral norms also we show that for every separable banach space containing l 1 and for every e 0 there is an equivalent norm so that every convex combination of w slices in the dual unit ball has diameter at least 2 e

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

octahedral norms in spaces of operators abstract we study octahedral norms in the space of bounded linear operators between banach spaces in fact we prove that l x y has octahedral norm whenever x and y have octahedral norm as a consequence the space of operators l l 1 x has octahedral norm if and only if x has octahedral norm these results also allow us to get the stability of strong diameter 2 property for projective tensor products of banach spaces which is an improvement of the known results about the size of nonempty relatively weakly open subsets in the unit ball of the projective tensor product of banach spaces

unconditional almost squareness and applications to spaces of lipschitz functions abstract we introduce an unconditional concept of almost squareness in order to provide a partial negative answer to the problem of existence of almost square dual banach spaces we also take advantage of this notion to provide some criteria of non duality of some subspaces of scalar as well as vector valued lipschitz functions

thin sets of constant width abstract we prove that every banach space which admits an unconditional basis can be renormed to contain a constant width set with empty interior thus guaranteeing for the first time existence of such sets in a reflexive space in the isometric case we prove that normal structure is characterized by the property that the class of diametrically complete sets and the class of sets with constant radius from the boundary coincide

a non uniform berry esseen bound via stein s method this paper is part of our efforts to develop stein s method beyond uniform bounds in normal approximation our main result is a proof for a non uniform berry esseen bound for independent and not necessarily identically distributed random variables without assuming the existence of third moments it is proved by combining truncation with stein s method and by taking the concentration inequality approach improved and adapted for non uniform bounds to illustrate the technique we give a proof for a uniform berry esseen bound without assuming the existence of third moments

w1 p quasiconvexity and variational problems for multiple integrals abstract variational problems for the multiple integral i u g u x dx where r m and u r n are studied a new condition on g called w1 p quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of c b morrey it being shown in particular to be necessary for sequential weak lower semicontinuity of i in w 1 p r n and for the existence of minimizers for certain related integrals counterexamples are given concerning the weak continuity properties of jacobians in w 1 p r n p n m an existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses

general existence theorems for hamilton jacobi equations in the scalar and vectorial cases keywords dirichlet problem hamilton jacobi equation existence of solutions reference caa article 1997 002doi 10 1007 bf02392708view record in web of science record created on 2008 11 25 modified on 2017 05 12

stein s method and a quantitative lindeberg clt for the fourier transforms of random vectors abstract we use a multivariate version of stein s method to establish a quantitative lindeberg clt for the fourier transforms of random n vectors we achieve this conceptually mainly by constructing a natural approach structure on n random vectors overlying the topology of weak convergence and technically mainly by deducing a specific integral representation for the hessian matrix of a solution to the stein equation with test function e t x exp i k 1 n t k x k where t x r n

on ideals of compact operators satisfying them r s inequality abstract letr s 0 1 we prove that a banach spacexsatisfies them r s inequality i e formula where is the canonical projection ofx ontox wheneverr s 2 1 and there exists a norm one projectionpon l x with ker p k x satisfying formula we characterize the last property of k x in l x by a strong version of the metric compact approximation property ofx our results extend some well known results onm ideals

big slices versus big relatively weakly open subsets in banach spaces abstract we study the unknown differences between the size of slices and relatively weakly open subsets of the unit ball in banach spaces we show that every banach space containing c 0 isomorphically can be equivalently renormed so that every slice of its unit ball has diameter 2 and still its unit ball contains nonempty relatively weakly open subsets with diameter arbitrarily small which answers an open question and stresses the differences between the size of slices and relatively weakly open subsets of the unit ball of banach spaces

an optimization algorithm applied to the morrey conjecture in nonlinear elasticity abstract for a long time it has been studied whether rank one convexity and quasiconvexity give rise to different families of constitutive relations in planar nonlinear elasticity stated in 1952 the morrey conjecture says that these families are different but no example has come forward to prove it now we attack this problem by deriving a specialized optimization algorithm based on two ingredients first a recently found necessary condition for the quasiconvexity of fourth degree polynomials that distinguishes between both classes in the three dimensional case and secondly upon a characterization of rank one convex fourth degree polynomials in terms of infinitely many constraints after extensive computational experiments with the algorithm we believe that in the planar case the necessary condition mentioned above is also necessary for the rank one convexity of fourth degree polynomials hence the question remains open

extreme differences between weakly open subsets and convex combinations of slices in banach spaces abstract we show that every banach space containing isomorphic copies of c 0 can be equivalently renormed so that every nonempty relatively weakly open subset of its unit ball has diameter 2 and however its unit ball still contains convex combinations of slices with diameter arbitrarily small which improves in an optimal way the known results about the size of this kind of subsets in banach spaces

arens regularity of banach algebras and the geometry of banach spaces we show that a banach algebra a such that a has the property v of a pelczynski is arens regular a strong uniqueness property of the extension of the product on a unital c algebra a to a is proved the algebraic structure of the bidual of a c algebra can be obtained through the local reflexivity principle we give examples which show that the results are sharp

existence of minimizers for non quasiconvex integrals keywords quasiconvexity existence of minimizers reference caa article 1995 002doi 10 1007 bf00380915 record created on 2008 11 25 modified on 2017 05 12

normal approximation under local dependence we establish both uniform and nonuniform error bounds of the berry esseen type in normal approximation under local dependence these results are of an order close to the best possible if not best possible they are more general or sharper than many existing ones in the literature the proofs couple stein s method with the concentration inequality approach

symmetric strong diameter two property in tensor products of banach spaces abstract we continue the investigation of the behavior of diameter two properties in tensor products of banach spaces our main result shows that the symmetric strong diameter two property is stable by taking projective tensor products we also prove a result for the symmetric strong diameter two property for the injective tensor product

convex values and lipschitz behavior of the complete hull mapping this note continues the study initiated in 2006 by p l papini r r phelps and the author on some classical notions from finite dimensional convex geometry in spaces of continuous functions let h be the family of all closed convex and bounded subsets of a banach space endowed with the hausdorff metric a completion of a h is a diametrically maximal set d h satisfying a c d and diam a diamd the complete hull mapping associates with every a h the family a of all its possible completions it is shown that the set valued mapping need not be convex valued even in finite dimensional spaces while in the case of c k spaces is convex valued if and only if k is extremally disconnected regarding the continuity we prove that again in c k spaces is always lipschitz continuous with constant less than or equal to 5 and has a lipschitz selection with constant less than or equal to 3 if we consider the analogous problem in euclidean spaces we show that is holder continuous of order 1 4 and locally holder continuous of order 1 2 the holder constants depending on the diameter of the sets in both cases

weakly open sets in the unit ball of the projective tensor product of banach spaces abstract a banach space is said to have the diameter two property if every non empty relatively weakly open subset of its unit ball has diameter two we prove that the projective tensor product of two banach spaces whose centralizer is infinite dimensional has the diameter two property the same statement also holds for x y if the centralizer of x is infinite dimensional and the unit sphere of y contains an element of numerical index one we provide examples of classical banach spaces satisfying the assumptions of the results if k is any infinite compact hausdorff topological space then c k y has the diameter two property for any nonzero banach space y we also provide a result on the diameter two property for the injective tensor product

some examples of rank one convex functions in dimension two keywords sobolev space quasiconvexity rank one convexity reference caa article 1990 004 record created on 2008 11 25 modified on 2017 05 12

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

an example of a quasiconvex function that is not polyconvex in two dimensions we study the different notions of convexity for the function f 2 2 2 det where e 2 2 introduced by dacorogna marcellini we show that f is convex polyconvex quasiconvex rank one convex if and only if 2 3 2 1 1 for some 0 2 3 respectively

fraisse limits in functional analysis we provide a unified approach to fraisse limits in functional analysis including the gurarij space the poulsen simplex and their noncommutative analogs we obtain in this general framework many known and new results about the gurarij space and the poulsen simplex and at the same time establish their noncommutative analogs particularly we construct noncommutative analogs of universal operators in the sense of rota

quasiconvex functions with subquadratic growth we establish the existence of quasiconvex functions which are not convex and which have subquadratic grow that infinity

central limit theorems for solutions of the kac equation speed of approach to equilibrium in weak metrics this paper is part of our efforts to show how direct application of probabilistic methods pertaining to central limit general theory can enlighten us about the convergence to equilibrium of the solutions of the kac equation here we consider convergence with respect to the following metrics kolmogorov s uniform metric 1 and 2 gini s dissimilarity indices widely known as 1 and 2 wasserstein metrics weighted metrics our main results provide new bounds or improvements on already well known ones for the corresponding distances between the solution of the kac equation and the limiting gaussian maxwellian distribution the study is conducted both under the necessary assumption that initial data have finite energy without assuming existence of moments of order greater than 2 and under the condition that the 2 moment of the initial distribution is finite for some 0

almost square banach spaces abstract we single out and study a natural class of banach spaces almost square banach spaces in an almost square space we can find given a finite set x 1 x 2 x n in the unit sphere a unit vector y such that x i y is almost one these spaces have duals that are octahedral and finite convex combinations of slices of the unit ball of an almost square space have diameter 2 we provide several examples and characterizations of almost square spaces we prove that non reflexive spaces which are m ideals in their biduals are almost square we show that every separable space containing a copy of c 0 can be renormed to be almost square a local and a weak version of almost square spaces are also studied

from stein identities to moderate deviations stein s method is applied to obtain a general cramer type moderate deviation result for dependent random variables whose dependence is defined in terms of a stein identity a corollary for zero bias coupling is deduced the result is also applied to a combinatorial central limit theorem a general system of binary codes the anti voter model on a complete graph and the curie weiss model a general moderate deviation result for independent random variables is also proved

weakly open sets in the unit ball of some banach spaces and the centralizer abstract we show that every banach space x whose centralizer is infinite dimensional satisfies that every non empty weakly open set in b y has diameter 2 where y n s x n fold symmetric projective tensor product of x endowed with the symmetric projective norm for every natural number n we provide examples where the above conclusion holds that includes some spaces of operators and infinite dimensional c algebras we also prove that every non empty weak open set in the unit ball of the space of n homogeneous and integral polynomials on x has diameter two for every natural number n whenever the cunningham algebra of x is infinite dimensional here we consider the space of n homogeneous integral polynomials as the dual of the space n s e x n fold symmetric injective tensor product of x endowed with the symmetric injective norm for instance every infinite dimensional l 1 satisfies that its cunningham algebra is infinite dimensional we obtain the same result for every non reflexive l embedded space and so for every predual of an infinite dimensional von neumann algebra

on the dual of cesaro function space abstract the goal of this paper is to present an isometric representation of the dual space to cesaro function space c p w 1 p induced by arbitrary positive weight function w on interval 0 l where 0 l for this purpose given a strictly decreasing nonnegative function on 0 l the notion of essential concave majorant f of a measurable function f is introduced and investigated as applications it is shown that every slice of the unit ball of the cesaro function space has diameter 2 consequently cesaro function spaces do not have the radon nikodym property are not locally uniformly convex and they are not dual spaces

rank one convexity implies polyconvexity for isotropic objective and isochoric elastic energies in the two dimensional case we show that in the two dimensional case every objective isotropic and isochoric energy function which is rank one convex on gl 2 is already polyconvex on gl 2 thus we negatively answer morrey s conjecture in the subclass of isochoric nonlinear energies since polyconvexity implies quasiconvexity our methods are based on different representation formulae for objective and isotropic functions in general as well as for isochoric functions in particular we also state criteria for these convexity conditions in terms of the deviatoric part of the logarithmic strain tensor

diametrically complete sets and normal structure abstract we prove that in some classes of reflexive banach spaces every maximal diametral set must be diametrically complete thus showing that diametrically complete sets may have empty interior also in reflexive spaces as a consequence we prove that in those spaces normal structure is equivalent to the weaker property that for every bounded set the absolute chebyshev radius is strictly smaller than the diameter moreover we prove that in any normed space the two classes of diametrically complete sets and of sets of constant radius from the boundary coincide if the space has normal structure

cauchy dirichlet problem for first order nonlinear systems abstract let r nbe open u r mand thus the gradient matrixdu r m n we lete r m nbe compact and denote byrcoeandpcoethe rank one convex and polyconvex hull ofe respectively we show that ifrcoe pcoe and two other hypotheses named the segment property and the extreme points property and if c1 r m is such that d x e int pcoe x then there exists a dense set of u w1 r m such that formula we apply this existence theorem to some relevant examples studied in the literature as well as to problems with x u dependence

relaxation results for functions depending on polynomials changing sign on rank one matrices abstract in this paper we are interested in computing the different convex envelopes of functions depending on polynomials especially those having it is main part change sign on rank one matrices our main result applies to functions of the type w f p f w f p f f det f or w f p f g adj n f defined on the space of matrices where f r r and g r 3 r are three continuous functions and p p 0 p 1 p d is a polynomial such that p d has the property of changing sign on rank one matrices then the polyconvex quasi convex and rank one convex envelopes of w are equal

banach spaces which are m ideals in their biduals we investigate banach spaces x such that x is an m ideal in x subspaces quotients and co sums of spaces which are m ideals in their biduals are again of this type a nonreflexive space x which is an m ideal in x contains a copy of co recently lima has shown that if k x is an m ideal in lf x then x is an m ideal in x here we show that if xis reflexive and k x is an m ideal in l x then k x is isometric to l x i e k x is an m ideal in its bidual moreover for real such spaces we show that k x contains a proper m ideal if and only if x or x contains a proper m ideal in 3 we start to study the properties of spaces which are m ideals in their biduals we show that subspaces and quotients of spaces which are m ideals in their biduals have the same property the proof of this result is based upon the fact that x is an m ideal in x if and only if the natural projection from x onto x is an l projection using local reflexivity we show that if x is an m ideal in x and x is nonreflexive then x contains almost isometric copies of co from this it follows that subspaces and quotients of such papers which are isomorphic to dual spaces are reflexive m structure of spaces which are m ideals in their biduals are studied in 4 here we show that if x is an m ideal in x without proper m summands and j o is an m ideal in x then x c j in 5 we treat the relation between spaces which are m ideals in their biduals and spaces of compact operators in 12 lima proved that if k x is an m ideal in lf x then x is an m ideal in x and x has the rnp here we show that if x is reflexive and k x is an m ideal in l x then k x is isometric to l x i e k x is an m ideal in its bidual moreover if k x is an m ideal in its bidual then x is reflexive we prove that for real spaces k x y contains a proper m summand