bidual renormings of banach spaces abstract given a separable banach space x with no isomorphic copies of l 1 and a separable subspace y of its bidual we provide a sufficient condition on y to ensure that x admits an equivalent norm such that the restriction to y of the corresponding bidual norm is midpoint locally uniformly rotund this result applies to the separable subspaces of the bidual of a banach space with a shrinking unconditional schauder basis and to the bidual of the james space

on the choice of norms in system identification in this paper we discuss smooth and sensitive norms for prediction error system identification when the disturbances are magnitude bounded formal conditions for sensitive norms which give an order of magnitude faster convergence of the parameter estimate variance are developed however it also is shown that the parameter estimate variance convergence rate of sensitive norms is arbitrarily bad for certain distributions a necessary condition for a norm to be statistically robust with respect to the family f c of distributions with support c c for some arbitrary c 0 is that its second derivative does not vanish on the support a direct consequence of this observation is that the quadratic norm is statistically robust among all l sub p norms p spl les 2 spl infin for f c

on a class of compactly epi lipschitzian sets the paper is devoted to the study of the so called compactly epi lipschitzian sets these sets are needed for many aspects of generalized differentiation particulary for necessary optimality conditions stability of mathematical programming problems and calculus rules for subdifferentials and normal cones we present general conditions under which sets defined by general constraints are compactly epi lipschitzian this allows us to show how the compact epi lipschitzness properties behave under set intersections

a maximum principle for nonsmooth optimal control problems with state constraints abstract optimality conditions are derived in the form of a maximum principle governing solutions to an optimal control problem which involves state constraints the conditions which apply in the absence of differentiability assumptions on the data are stated in terms of clarke s generalized jacobians although not the most general available the conditions are derived by a novel method this involves removal of the state constraints by introduction of a penalty term and application of ekeland s variational principle

on a classical renorming construction of v klee we further develop a classical geometric construction of v klee and show typically that if x is a nonreflexive banach space with separable dual then x admits an equivalent norm which is frechet differentiable locally uniformly rotund its dual norm is uniformly g teaux differentiable the weak and the norm topologies coincide on the sphere of x and yet is not rotund this proves a stronger form of a conjecture of v klee

strictly convex norms and topology we introduce a new topological property called and the corresponding class of topological spaces which includes spaces with g diagonals and gruenhage spaces using we characterize those banach spaces which admit equivalent strictly convex norms and give an internal topological characterization of those scattered compact spaces k for which the dual banach space c k admits an equivalent strictly convex dual norm we establish some relationships between and other topological concepts and the position of several well known examples in this context for instance we show that c k admits an equivalent strictly convex dual norm where k is kunen s compact s space also under additional axioms we provide examples of compact scattered non gruenhage spaces of cardinality 1 having

the fermat rule for multifunctions on banach spaces using variational analysis we study vector optimization problems with objectives being closed multifunctions on banach spaces or in asplund spaces in particular in terms of the coderivatives we present fermat s rules as necessary conditions for an optimal solution of the above problems as applications we also provide some necessary conditions in terms of clarke s normal cones or the limiting normal cones for pareto efficient points

continuous functions on totally ordered spaces that are compact in their order topologies abstract the totally ordered spaces that are compact and connected in their order topologies are characterized as images of lexicographic cubes the banach space of continuous functions on the lexicographic product of compact totally ordered spaces whose spaces of continuous functions have locally uniformly convex norms has an equivalent locally uniformly convex norm if and only if the product is countable the banach space of continuous functions on a compact totally ordered space always has an equivalent kadec norm those compact totally ordered spaces for which the banach space of continuous functions has an equivalent locally uniformly convex norm are characterized in terms of the bounded decreasing interval functions that can be defined on the intervals of the totally ordered spaces some examples are discussed in detail

topological properties of banach spaces weakly compact sets in banach spaces are fragmentable we introduce the weaker notion of the a fragmentability of a subset of a banach space a notion that has a far wider range of applicability we prove that all weakly cech analytic subsets of a banach space are a fragmented this implies that all banach spaces having kadec norms are a fragmented however l is not fragmented

the non regular ceo problem we consider the ceo problem for non regular source distributions such as uniform or truncated gaussian a group of agents observe independently corrupted versions of data and transmit coded versions over rate limited links to a ceo the ceo then estimates the underlying data based on the received coded observations agents are not allowed to convene before transmitting their observations this formulation is motivated by the practical problem of a firm s ceo estimating non regular beliefs about a sequence of events before acting on them agents observations are modeled as jointly distributed with the underlying data through a given conditional probability density function we study the asymptotic behavior of the minimum achievable mean squared error distortion at the ceo in the limit when the number of agents l and the sum rate r tend to infinity we establish a 1 r 2 convergence of the distortion an intermediate regime of performance between the exponential behavior in discrete ceo problems berger zhang and viswanathan 1996 and the 1 r behavior in gaussian ceo problems viswanathan and berger 1997 achievability is proved by a layered architecture with scalar quantization distributed entropy coding and midrange estimation the converse is proved using the bayesian chazan zakai ziv bound

alur dual renormings of banach spaces we give a covering type characterization for the class of dual banach spaces with an equivalent alur dual norm

the fuzzy intersection rule in variational analysis with applications the fuzzy intersection rule for frechet normal cones in asplund spaces was established by mordukhovich and the author using the extremal principle which appears more convenient to apply in some applications in this paper we present a complete discussion of this rule in various aspects we show that the fuzzy intersection rule is another characterization of the asplund property of the space various applications are considered as well in particular a complete set of fuzzy calculus rules for general lower semicontinuous functions are established

robust identification and galois sequences abstract worst case l1 identification is studied for bibo stable linear shift invariant systems it is shown that the chebyshev identification method when used with galois input designs satisfies a certain robust convergence property and provides l1 model error bounds in worst case identification of bibo stable systems with a uniformly bounded noise set up the robust identification methodology developed is compatible with the modelling requirements of modern robust control design

on the fuzzy intersection rule abstract the paper provides a complete exposition of the fuzzy intersection rule in variational analysis and its applications to generalized differentiation theory new forms of the rule and further applications are presented

metric regularity tangential distances and generalized differentiation in banach spaces abstract in this paper we establish new generalized differentiation rules in general banach spaces regarding normal cones to set images under functions coderivatives of compositions of set valued mappings as well as calculus results for normal compactness of sets and their images in addition to the metric regularity of mappings our results involve tangential distances of sets for which we also provide a fairly complete study by exploring its variations basic properties as well as relations to similar notions some related results are also established

smoothing of bump functions abstract let x be a separable banach space with a schauder basis admitting a continuous bump which depends locally on finitely many coordinates then x admits also a c smooth bump which depends locally on finitely many coordinates

the size of the membership set in a probabilistic framework in this paper we study the size of the membership set for system identification in a probabilistic framework assuming that the regressors are persistently exciting and the measurement noise is a sequence of independent identically distributed bounded random variables lower and upper non asymptotic probability bounds on the membership set diameter are obtained these bounds are used in the computation of the confidence intervals for interpolatory estimators

generalized sequential normal compactness in banach spaces abstract normal compactness conditions are important properties of sets set valued mappings in variational analysis which are generalized versions of the classical lipschitzian property and are essential for the calculus of generalized differentiation theory in this paper we propose the notion called the generalized sequential normal compactness and establish its basic properties and calculus in general banach spaces

convexity and generalized second order derivatives for locally lipschitz functions abstract we define the generalized second order directional derivatives by means of the clarke generalized gradient for locally lipschitz functions then we give characterization of convexity and state a new sufficient optimality condition

differentiation of sets in measure abstract suppose f e for each e 0 1 is a bounded borel subset of r d and f e f 0 as e 0 let a e f e f 0 be symmetric difference and p be an absolutely continuous measure on r d we introduce the notion of derivative of f e with respect to e d f e d e d a e d e such that d d e p a e e 0 q d d e a e e 0 where q is another explicitly described measure although not in r d we discuss why this sort of derivative is needed to study local point processes in neighbourhood of a set in short if sequence of point processes n n n 1 2 is given on the class of set valued mappings f f such that all f e converge to the same f f 0 then the weak limit of the local processes n n a e f e f lives on the class of derivative sets d f e d e e 0 f f we compare this notion of the derivative set valued mapping with other existing notions

convex subcones of the contingent cone in nonsmooth calculus and optimization the tangential approximants most useful in nonsmooth analysis and optimization are those which lie between the clarke tangent cone and the bouligand contigent cone a study of this class of tangent cones is undertaken here it is shown that although no convex subcone of the contingent cone has the isotonicity property of the contingent cone there are such convex subcones which are more accurate approximants than the clarke tangent cone and possess an associated subdifferential calculus that is equally strong in addition a large class of convex subcones of the contingent cone can replace the clarke tangent cone in necessary optimality conditions for a nonsmooth mathematical program however the clarke tangent cone plays an essential role in the hypotheses under which these calculus rules and optimality conditions are proven overall the results obtained here suggest that the most complete theory of nonsmooth analysis combines a number of different tangent cones

epi lipschitz like sets in banach space theorems and examples on considere un espace de banach x et on etudie quelles proprietes un ensemble c dans x doit avoir pour que le cone tangent de clarke t c x et le cone normal n c x mesurent adequatement le comportement a la frontiere de x dans c

on trees and dual rotund norms abstract we give a necessary and sufficient condition for the existence of an equivalent dual rotund norm on c 0 l 1 where is a tree the condition is expressed succinctly in terms of the embeddability of into a particular totally ordered set and compares very well with the analogous situation for local uniform rotundity this resolves an open problem from haydon s work in asplund spaces trees and renorming theory

on binormality in non separable banach spaces we study binormality a separation property of the norm and weak topologies of a banach space we show that every banach space which belongs to a p class is binormal we also show that the asplundness of a banach space is equivalent to a related separation property of its dual space

constrained logarithmic least squares in parameter estimation this paper first shows that while robust in terms of the average output error the least squares estimate is sensitive to outliers with respect to the maximum output error in fact the worst case output error of the least squares can go unbounded then a constrained logarithmic least squares for system identification is proposed analytic center algorithms are presented to solve this constrained logarithmic least squares problem

variational analysis in nonsmooth optimization and discrete optimal control this paper is devoted to applications of modern methods of variational analysis to constrained optimization and control problems generally formulated in infinite dimensional spaces the main focus is on the study of problems with nonsmooth structures which require the usage of advanced tools of generalized differentiation in this way we derive new necessary optimality conditions in optimization problems with functional and operator constraints and then apply them to optimal control problems governed by discrete time inclusions in infinite dimensions the principal difference between finite dimensional and infinite dimensional frameworks of optimization and control consists of the lack of compactness in infinite dimensions which leads to imposing certain normal compactness properties and developing their comprehensive calculus together with appropriate calculus rules of generalized differentiation on the other hand one of the most important achievements of the paper consists of relaxing the latter assumptions for certain classes of optimization and control problems in particular we fully avoid the requirements of this type imposed on target endpoint sets in infinite dimensional optimal control for discrete time inclusions

layer potential methods for elliptic homogenization problems in this paper we use the method of layer potentials to study l 2 boundary value problems in a bounded lipschitz domain for a family of second order elliptic systems with rapidly oscillating periodic coefficients defining l e div a e 1 x under the assumption that a x is elliptic symmetric periodic and holder continuous we establish the solvability of the l 2 dirichlet regularity and neumann problems for l e u e 0 in with optimal estimates uniform in e 0

polyhedral norms on non separable banach spaces abstract we prove the existence of equivalent polyhedral norms on a number of classes of non separable spaces the majority of which being of the form c k in particular we obtain a complete characterization of those trees t such that c 0 t admits an equivalent polyhedral norm

on weakly locally uniformly rotund banach spaces abstract we show that every normed space e with a weakly locally uniformly rotund norm has an equivalent locally uniformly rotund norm after obtaining a discrete network of the unit sphere s e for the weak topology we deduce that the space e must have a countable cover by sets of small local diameter which in turn implies the renorming conclusion this solves a question posed by deville godefroy haydon and zizler for a weakly uniformly rotund norm we prove that the unit sphere is always metrizable for the weak topology despite the fact that it may not have the kadec property moreover banach spaces having a countable cover by sets of small local diameter coincide with the descriptive banach spaces studied by hansell so we present here some new characterizations of them

convergence analysis of stationary points in sample average approximation of stochastic programs with second order stochastic dominance constraints sample average approximation saa method has recently been applied to solve stochastic programs with second order stochastic dominance ssd constraints in particular hu et al math program 133 171 201 2012 presented a detailed convergence analysis of epsilon optimal values and epsilon optimal solutions of sample average approximated stochastic programs with polyhedral ssd constraints in this paper we complement the existing research by presenting convergence analysis of stationary points when saa is applied to a class of stochastic minimization problems with ssd constraints specifically under some moderate conditions we prove that optimal solutions and stationary points obtained from solving sample average approximated problems converge with probability one to their true counterparts moreover by exploiting some recent results on large deviation of random functions and sensitivity analysis of generalized equations we derive exponential rate of convergence of stationary points

infinite dimensional generalized jacobian properties and calculus rules the extension to infinite dimensional domains of clarke s generalized jacobian is the focus of this paper first a generalization of a fabian preiss theorem to the infinite dimensional setting is obtained as a consequence a new formula relating the clarke s generalized jacobians corresponding to finite dimensional spaces k l with k l is established furthermore in the infinite dimensional case basic properties pertaining the generalized jacobian are developed and then an identification of this set valued map is produced applications of these results in the form of chain rules including sum and product rules and a computational formula for continuous selections are derived

weak locally uniformly rotund norms and descriptive compact spaces we prove that a dual banach space x has an equivalent w lur norm if and only if the weak topology has a isolated network we give sufficient conditions for the existence of equivalent norms with various kinds of differentiability

epi lipschitzian reachable sets of differential inclusions abstract the reachable sets of a differential inclusion have nonsmooth topological boundaries in general the main result of this paper is that under the well known assumptions of filippov s existence theorem about solutions of differential inclusions every epi lipschitzian initial compact set k r n preserves this regularity for a short time i e f t k is also epi lipschitzian for all small t 0 the proof is based on rockafellar s geometric characterization of epi lipschitzian sets and uses a new result about the inner semicontinuity of clarke tangent cone t y t f t k c y r n with respect to both arguments

parabolic regularity and dirichlet boundary value problems abstract we study the relationship between the regularity and dirichlet boundary value problems for parabolic equations of the form l u div a u u t 0 in lip 1 1 2 time varying cylinders where the coefficient matrix a a i j x t is uniformly elliptic and bounded we show that if the regularity problem r p for the equation l u 0 is solvable for some 1 p then the dirichlet problem d p for the adjoint equation l v 0 is also solvable where p p p 1 this result is an analogue of the result established in the elliptic case by kenig and pipher 1993 in the parabolic settings in the special case of the heat equation in slightly smoother domains this has been established by hofmann and lewis 1996 and nystrom 2006 for scalar parabolic systems in comparison our result is abstract with no assumption on the coefficients beyond the ellipticity condition and is valid in more general class of domains

l2 solvability of the dirichlet neumann and regularity problems for parabolic equations with time independent holder continuous coefficients we establish the l 2 solvability of dirichlet neumann and regularity problems for divergence form heat or diffusion equations with timeindependent holder continuous diffusion coefficients on bou

locally uniformly convex norms in banach spaces and their duals abstract it is shown that a banach space with locally uniformly convex dual admits an equivalent norm that is itself locally uniformly convex

sequential normal compactness versus topological normal compactness in variational analysis abstract we study relationships between two normal compactness properties of sets in banach spaces that play an essential role in many aspects of variational analysis and its applications particularly in calculus rules for generalized differentiation necessary optimality and suboptimality conditions for optimization problems etc both properties automatically hold in finite dimensional spaces and reveal principal features of the infinite dimensional variational theory similar formulations of these properties involve the weak convergence of sequences and nets respectively containing generalized normal cones in duals to banach spaces we prove that these properties agree for a large class of banach spaces that include weakly compactly generated spaces we also show that they are always different in banach spaces whose unit dual ball is not weak sequentially compact moreover the sequential and topological normal compactness properties may not coincide even in non separable asplund spaces that admit an equivalent c smooth norm

a smooth variational principle with applications to hamilton jacobi equations in infinite dimensions abstract we prove that if x is a banach space which admits a smooth lipschitzian bump function then for every lower semicontinuous bounded below function there exists a lipschitzian smooth function g on x such that f g attains its strong minimum on x thus extending a result of borwein and preiss we then show how the above result can be used to obtain existence and uniqueness results of viscosity solutions of hamilton jacobi equations in infinite dimensional banach spaces a without assuming the radon nikodym property

the hypertangent cone in borwein and strojwas 5 we have observed that in baire metrizable spaces the hypertangent cone plays an important role in the theory of tangent cones and generalized subgradients because its properties relate to lipschitz behaviour of sets and functions here more accurate results for banach spaces are presented they follow from a formula which may be viewed as the discrete version of the treiman inclusion 18 we prove this in section 1 our discrete formula has many applications it implies generalizations of the bishop phelps theorem on the density of support points 11 and the dense clarke subdifferentiability theorem of mclinden 14 in section 2 we present a new characterization of the clarke derivative and new conditions characterizing directionally lipschitzian functions on a banach space in section 3 we use our discrete formula to obtain mean value theorems for lower semicontinuous functions on banach spaces in the vein of penot 16 also we prove a strengthening of lebourg s 13 mean value theorem for reflexive spaces in section 4 we prove a general inversion theorem in particular we use it to furnish conditions for the clarke tangent cone of the intersection of sets to contain the intersection of the tangent cones for these sets this generalizes the finite dimensional version of ioffe ll and an epi lipschitzian version of rockafellar 16 in a banach space the basic facts about tangent cones and generalized subgradients that we use in this paper are discussed in clarke l0

optimally robust system identification of systems subject to amplitude bounded stochastic disturbances in this paper it is shown that log cos spl pi x 2c is the optimally robust criterion function for prediction error methods with respect to amplitude bounded stochastic disturbances this criterion function minimizes the maximum asymptotic covariance matrix of the parameter estimates for the family of innovations of the systems which are amplitude bounded by the constant c furthermore the stochastic worst case performance of the estimate corresponding to the criterion function log cos spl pi x 2c is better than the worst case performance of the least squares estimate even if the constant c is chosen larger than the actual amplitude bound on the innovations in addition to its favorable properties in a stochastic setting this criterion function also generates estimates which are unfalsified in a deterministic framework

differentiability of lipschitz functions on banach spaces abstract the main result of this note says that if the norm of a banach space e is differentiable frechet gateaux or in some intermediate sense away from the origin then every locally lipschitz function defined on an open subset g of e is differentiable in the same sense at every point of some subset of g dense in g as the main corollary we prove that every lipschitz function on an asplund space is frechet differentiable on a dense set moreover in all these results the mean value theorem for the points of differentiability which is equivalent to the possibility of computing the clarke subdifferential from the points of differentiability is shown to hold

variational analysis of extended generalized equations via coderivative calculus in asplund spaces abstract this paper is devoted to the development of variational analysis and generalized differentiation in the framework of asplund spaces we mainly concern the study of a special class of set valued mapping given in the form s x y y 0 f x y q x y x x where both f and q are set valued mappings between asplund spaces models of this type are associated with solutions maps to the so called extended generalized equations and play a significant role in many aspects of variational analysis and its applications to optimization stability control theory etc in this paper we conduct a local variational analysis of such extended solution maps s and their remarkable specifications based on dual space generalized differential constructions of the coderivative type the major part of our analysis revolves around coderivative calculus largely developed and implemented in this paper and then applied to establishing verifiable conditions for robust lipschitzian stability of extended generalized equations and related objects

some remarks on smooth renormings of banach spaces abstract we prove that in every separable banach space x with a schauder basis and a c k smooth norm it is possible to approximate uniformly on bounded sets every equivalent norm with a c k smooth one in a way that the approximation is improving as fast as we wish on the elements depending only on the tail of the schauder basis our result solves a problem from the recent monograph of guirao montesinos and zizler

upward slopes and inf convolutions nonsmooth analysis and exact penalty methods give much prominence to functions that exit from certain lower level sets with positive upward slope a novel weak version of such slopes is formalized here and studied in terms of differential properties applied to inf convolutions those properties help to recover and extend recent results on nonconvex separation cooperative production or market games and welfare economics

links between directional derivatives through multidirectional mean value inequalities we prove in the general setting of lower semicontinuous functions on banach spaces the relation between the rockafellar directional derivative and the mixed lower limit of the lower dini derivatives as a byproduct we derive the famous inclusions of tangent cones of closed sets in banach spaces the results are established using as principal tool multidirectional mean value inequalities aussel et al siam j optim 9 3 690 706 1999

the lagrange multiplier rule for super efficiency in vector optimization in this paper we study constrained multiobjective optimization problems with objectives being closed graph multifunctions in banach spaces in terms of the coderivatives and clarke s normal cones we establish lagrange multiplier rules for super efficiency as necessary or sufficient optimality conditions of the above problems

brief new identification approaches for disturbed models recently much research has been conducted in the field of identification of the linear models in general these methods use a time domain estimate or a frequency domain estimate in this paper the time domain estimate and the frequency domain estimate were combined to identify the autoregressive exogenous noise arx interference model the concept of a general prediction error criterion is introduced for the time domain estimate an optimal frequency estimation is introduced for the frequency domain estimate a new identification method called the empirical frequency domain optimal parameter estimate is proposed for disturbed systems it is fully applied and developed for the output error model and a specific case or the arx model the algorithm theoretically provides the globally optimum frequency domain estimate of the model some simulations are included to illustrate the new identification method

average locally uniform rotundity and a class of nonlinear maps abstract we consider some topological characterizations of dual banach spaces that admit an equivalent dual average locally uniformly rotund norm and provide a criterion for such renorming which involves the class of slicely continuous maps

on strong tangential transversality abstract this is the second of two closely related papers on transversality here we introduce the notion of strong tangential transversality of two closed subsets of a banach space which is a natural sufficient condition for tangential transversality some properties of uniform tangent sets are obtained a new sum rule for the clarke subdifferential is proven

nonsmooth calculus in finite dimensions the notion of subgradient originally defined for convex functions has in recent years been extended via the upper subderivative to cover functions that are not necessarily convex or even continuous a number of calculus rules have been proven for these generalized subgradients this paper develops the finite dimensional generalized subdifferential calculus for strictly lower semicontinuous functions under considerably weaker hypotheses than those previously used the most general finite dimensional convex subdifferential calculus results are recovered as corollaries other corollaries given include new necessary conditions for optimality in a nonsmooth mathematical program various chain rule formulations are considered equality in the subdifferential calculus formulae is proven under hypotheses weaker than the usual subdifferential regularity assumptions

uniformly gâteaux smooth approximations on c0 γ abstract every lipschitz mapping from c 0 into a banach space y can be uniformly approximated by lipschitz mappings that are simultaneously uniformly g teaux smooth and c frechet smooth

powerful and flexible fuzzy algorithm for nonlinear dynamic system identification a new powerful and flexible fuzzy algorithm for nonlinear dynamic system identification is presented it is based on the identification of the derivative of the system state instead of the future system state the membership functions of the underlying static fuzzy model are two sided gaussian functions and the learning algorithm is a hybrid nested routine based on least squares quasi newton and simplex optimization methods moreover a simple clustering algorithm based on an additional higher level fuzzy model is proposed the application to the identification of the mackey glass chaotic time series is presented and compared with previous results in terms of maximum error and nondimensional error index finally the application to a test nonlinear dynamic system is presented to show the capabilities of the clustering algorithm the obtained results show that the proposed algorithm can find wide application in practical problems such as in nonlinear electronic circuit design

hilbert generated spaces we classify several classes of the subspaces of banach spaces x for which there is a bounded linear operator from a hilbert space onto a dense subset in x dually we provide optimal affine homeomorphisms from weak star dual unit balls onto weakly compact sets in hilbert spaces or in c0 spaces in their weak topology the existence of such embeddings is characterized by the existence of certain uniformly g teaux smooth norms

an extended extremal principle with applications to multiobjective optimization we develop an extended version of the extremal principle in variational analysis that can be treated as a variational counterpart to the classical separation results in the case of nonconvex sets and which plays an important role in the generalized differentiation theory and its applications to optimization related problems the main difference between the conventional extremal principle and the extended version developed below is that instead of the translation of sets involved in the extremal systems we allow deformations the new version seems to be more flexible in various applications and covers in particular multiobjective optimization problems with general preference relations in this way we obtain new necessary optimality conditions for constrained problems of multiobjective optimization with nonsmooth data and also for multiplayer multiobjective games

gruenhage compacta and strictly convex dual norms abstract we prove that if k is a gruenhage compact space then c k admits an equivalent strictly convex dual norm as a corollary we show that if x is a banach space and x span k where k is a gruenhage compact in the w topology and is equivalent to a coarser w lower semicontinuous norm on x then x admits an equivalent strictly convex dual norm we give a partial converse to the first result by showing that if is a tree then c 0 admits an equivalent strictly convex dual norm if and only if is a gruenhage space finally we present some stability properties satisfied by gruenhage spaces in particular gruenhage spaces are stable under perfect images