hard lefschetz theorem for valuations complex integral geometry and unitarily invariant valuations we obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets a version of the hard lefschetz theorem using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations it implies new integral geometric formulas for real submanifolds in hermitian spaces generalizing the classical kinematic formulas in euclidean

even minkowski valuations a new integral representation of smooth translation invariant and rotation equivariant even minkowski valuations is established explicit formulas relating previously obtained descriptions of such valuations with the new more accessible one are also derived moreover the action of alesker s hard lefschetz operators on these minkowski valuations is explored in detail

on the pointwise regularity of functions in critical besov spaces abstract we bound the spectrum of singularities of functions in the critical besov spaces and we show that this result is sharp in the sense that equality in the bounds holds for quasi every function of the corresponding besov space

regularity criteria for almost every function in sobolev spaces in this paper we determine the multifractal nature of almost every function in the prevalence setting in a given sobolev or besov space according to different regularity exponents these regularity criteria are based on local l p regularity or on wavelet coefficients and give a precise information on pointwise behavior

a characterization of affine surface area abstract we show that every upper semicontinuous and equi affine invariant valuation on the space of d dimensional convex bodies is a linear combination of affine surface area volume and the euler characteristic

sl m c equivariant and translation covariant continuous tensor valuations abstract the space of continuous sl m c equivariant m 2 and translation covariant valuations taking values in the space of real symmetric tensors on c m r 2 m of rank r 0 is completely described the classification involves the moment tensor valuation for r 1 and is analogous to the known classification of the corresponding tensor valuations that are sl 2 m r equivariant although the method of proof cannot be adapted

local behavior of traces of besov functions prevalent results let 1 d d and p q s satisfying 0 p 0 q 0 s d p in this article we study the global and local regularity properties of traces on affine subsets of rd of functions belonging to the besov space bp qs rd given a d dimensional subspace h rd for almost all functions in bp qs rd in the sense of prevalence we are able to compute the singularity spectrum of the traces fa of f on affine subspaces of the form a h for lebesgue almost every a rd d in particular we prove that for lebesgue almost every a rd d these traces fa are more regular than what could be expected from standard trace theorems and that fa enjoys a multifractal behavior

structure of the unitary valuation algebra s alesker has shown that if g is a compact subgroup of o n acting transitively on the unit sphere sn 1 then the vector space valg of continuous translation invariant g invariant convex valuations on rn has the structure of a finite dimensional graded algebra over r satisfying poincare duality we show that the kinematic formulas for g are determined by the product pairing using this result we then show that the algebra valu n is isomorphic to r s t fn 1 fn 2 where s t have degrees 2 and 1 respectively and the polynomial fi is the degree i term of the power series log 1 s t

valuations and busemann petty type problems projection and intersection bodies define continuous and gl n contravariant valuations they played a critical role in the solution of the shephard problem for projections of convex bodies and its dual version for sections the busemann petty problem we consider the question whether fkfl implies v k v l where f is a homogeneous continuous operator on convex or star bodies which is an so n equivariant valuation important previous results for projection and intersection bodies are extended to a large class of valuations

difference bodies in complex vector spaces abstract a complete classification is obtained of continuous translation invariant minkowski valuations on an m dimensional complex vector space which are covariant under the complex special linear group

generalized translation invariant valuations and the polytope algebra abstract we study the space of generalized translation invariant valuations on a finite dimensional vector space and construct a partial convolution which extends the convolution of smooth translation invariant valuations our main theorem is that mcmullen s polytope algebra is a subalgebra of the partial convolution algebra of generalized translation invariant valuations more precisely we show that the polytope algebra embeds injectively into the space of generalized translation invariant valuations and that for polytopes in general position the convolution is defined and corresponds to the product in the polytope algebra

prevalence a translation invariant almost every on infinite dimensional spaces we present a measure theoretic condition for a property to hold almost everywhere on an infinite dimensional vector space with particularemphasis on function spaces such as c k and l p like the concept of lebesgue almost every on an infinite dimensional spaces our notion of prevalence is translation invariant instead of using a specific measure on the entire space we define prevalence in terms of the class of all probability measures with compact support prevalence is a more appropriate condition than the topological concepts of open and dense or generic when one desires a probabilistic result on the likelihood of a given property on a function space

selfish routing in capacitated networks according to wardrop s first principle agents in a congested network choose their routes selfishly a behavior that is captured by the nash equilibrium of the underlying noncooperative game a nash equilibrium does not optimize any global criterion per se and so there is no apparent reason why it should be close to a solution of minimal total travel time i e the system optimum in this paper we offer positive results on the efficiency of nash equilibria in traffic networks in contrast to prior work we present results for networks with capacities and for latency functions that are nonconvex nondifferentiable and even discontinuous the inclusion of upper bounds on arc flows has early been recognized as an important means to provide a more accurate description of traffic flows in this more general model multiple nash equilibria may exist and an arbitrary equilibrium does not need to be nearly efficient nonetheless our main result shows that the best equilibrium is as efficient as in the model without capacities moreover this holds true for broader classes of travel cost functions than considered hitherto

intersection bodies and valuations all gl n covariant star body valued valuations on convex polytopes are completely classified it is shown that there is a unique nontrivial such valuation this valuation turns out to be the so called intersection operator an operator that played a critical role in the solution of the busemann petty problem

minkowski valuations under volume constraints abstract we provide a description of the space of continuous and translation invariant minkowski valuations k n k n for which there is an upper and a lower bound for the volume of k in terms of the volume of the convex body k itself although no invariance with respect to a group acting on the space of convex bodies is imposed we prove that only two types of operators appear a family of operators having only cylinders over n 1 dimensional convex bodies as images and a second family consisting essentially of 1 homogeneous operators using this description we give improvements of some known characterization results for the difference body

log concavity properties of minkowski valuations new orlicz brunn minkowski inequalities are established for rigid motion compatible minkowski valuations of arbitrary degree these extend classical log concavity properties of intrinsic volumes and generalize seminal results of lutwak and others two different approaches which refine previously employed techniques are explored it is shown that both lead to the same class of minkowski valuations for which these inequalities hold an appendix by semyon alesker contains the proof of a new classification of generalized translation invariant valuations

design of price mechanisms for network resource allocation via price of anarchy we study the design of price mechanisms for communication network problems in which a user s utility depends on the amount of flow she sends through the network and the congestion on each link depends on the total traffic flows over it the price mechanisms are characterized by a set of axioms that have been adopted in the cost sharing games and we search for the price mechanisms that provide the minimum price of anarchy we show that given the non decreasing and concave utilities of users and the convex quadratic congestion costs in each link if the price mechanism cannot depend on utility functions the best achievable price of anarchy is 4 3 2 sqrt 2 approx 31 4 thus the popular marginal cost pricing with price of anarchy less than 1 3 33 3 is nearly optimal we also investigate the scenario in which the price mechanisms can be made contingent on the users preference profile while such information is available

volume of mixed bodies by using inequalities obtained for the volume of mixed bodies and the petty projection inequality sharp isoperimetric inequalities are derived for the projection measures quermassintegrale of a convex body these projection measure inequalities which involve mixed projection bodies zonoids are shown to be strengthened versions of the classical inequalities between the projection measures of a convex body the inequality obtained for the volume of mixed bodies is also used to derive a form of the brunn minkowski inequality involving mixed bodies as an application inequalities are given between the projection measures of convex bodies and the mixed projection integrals of the bodies o introduction in 24 it was shown that the petty projection inequality 26 will yield extensions of the known isoperimetric inequalities between the volume of a convex body and its projection measures quermassintegrale in this article we show that the petty projection inequality can be used to obtain sharp isoperimetric inequalities corollary 5 20 which are strengthened forms of the classical inequalities between the projection measures of a convex body this is accomplished by using the so called mixed convex bodies introduced by firey 17 geometric inequalities are derived for the volume of mixed bodies theorems 4 1 and 4 3 which when combined with the petty projection inequality yield inequalities involving projection bodies zonoids which are related to the inequalities between the projection measures of a convex body the inequalities for mixed bodies are also used to obtain a version of the brunn minkowski inequality theorem 4 2 involving mixed bodies also derived are inequalities between the volume of convex bodies and their associated mixed projection integral theorems 6 4 and 6 4 that are extensions and improvements of some previously established inequalities since some of our results concern projection bodies a topic of interest to workers outside the area of geometric convexity we have restated a few basic definitions and standard results from the field of geometric inequalities in 1 we collect for quick reference some well known results concerning mixed volumes and mixed area measures which will be required later references for material presented in this section are bonnesen and fenchel 7 for mixed volumes busemann 9 and leichtweiss 20 the original works of aleksandrov 1 4 and fenchel and jessen 15 supplemented by the survey of schneider 28 are recommended references for mixed area measures 2 contains introductory material regarding blaschke addition received by the editors january 8 1985 1980 mcithematics subject classihecition primary 52a40 secondary 53a15 46b20

pricing with markups in industries with increasing marginal costs we study a game that models a market in which heterogeneous producers of perfect substitutes make pricing decisions in a first stage followed by consumers that select a producer that sells at lowest price as opposed to cournot or bertrand competition producers select prices using a supply function that maps prices to production levels solutions of this type of models are normally referred to as supply function equilibria we consider a market where producers convex costs functions are proportional to each other depending on the efficiency of each particular producer we provide necessary and sufficient conditions for the existence of an equilibrium that uses simple supply functions that replicate the cost structure we then specialize the model to monomial cost functions with exponent q 0 which allows us to reinterpret the simple supply functions as a markup applied to the production cost we prove that an equilibrium for the markups exists if and only if the number of producers in the market is strictly larger than 1 q and if an equilibrium exists it is unique the main result for monomials is that the equilibrium nearly minimizes the total production cost when the market is competitive the result holds because when there is enough competition markups are bounded thus preventing prices to be significantly distorted from costs focusing on the case of linear unit cost functions on the production quantities we characterize the equilibrium accurately and refine the previous result to establish an almost tight bound on the worst case inefficiency of equilibria finally we derive explicitly the producers best response for series parallel networks with linear unit cost functions extending our previous result to more general topologies we prove that a unique equilibrium exists if and only if the network that captures the market structure is 3 edge connected for non series parallel markets we provide an example that does not admit an equilibrium on markups

the complex busemann petty problem on sections of convex bodies abstract the complex busemann petty problem asks whether origin symmetric convex bodies in c n with smaller central hyperplane sections necessarily have smaller volume we prove that the answer is affirmative if n 3 and negative if n 4

contravariant minkowski valuations a complete classification of all continuous gl n contravariant minkowski valuations is established as an application we present a family of sharp isoperimetric inequalities for such valuations which generalize the classical petty projection inequality

inequalities for mixed projection bodies mixed projection bodies are related to ordinary projection bodies zonoids in the same way that mixed volumes are related to ordinary volume analogs of the classical inequalities from the brunn minkowski theory such as the minkowski brunn minkowski and aleksandrov fenchel inequalities are developed for projection and mixed projection bodies

crofton measures and minkowski valuations a description of continuous rigid motion compatible minkowski valuations is established as an application we present a brunn minkowski type inequality for intrinsic volumes of these valuations

projection functions area measures and the alesker fourier transform abstract dual to koldobsky s notion of j intersection bodies the class of j projection bodies is introduced generalizing minkowski s classical notion of projection bodies of convex bodies a fourier analytic characterization of j projection bodies in terms of their area measures of order j is obtained in turn this yields an equivalent characterization of j projection bodies involving alesker s fourier type transform on translation invariant smooth spherical valuations as applications of these results several basic properties of j projection bodies are established and new non trivial examples are constructed

local dimensions of sliced measures and stability of packing dimensions of sections of sets abstract let m and n be integers with 0 r n to certain properties of plane sections of this leads us to prove among other things that the lower local dimension of n m plane sections of is typically constant provided that the hausdorff dimension of is greater than m the analogous result holds for the upper local dimension if has finite t energy for some t m we also give a sufficient condition for stability of packing dimensions of section of sets

even valuations on convex bodies the notion of eveil valuation is introduced as a natural generalization of volume on cornpact convex subsets of euclidean space a recent characterization theorem for volume leads in turn to a connection between even valuations on compact convex sets and continuous functions on grassmannians this connection can be described in part using generating distributions for symmetric compact convex sets we also explore some consequences of these characterization results in convex and integral geometry recent interest in volume as a valuation on compact conlvex sets stems from hilbert s third problem which is actually an ancient problem recast in modern terms hilbert asked if two polytopes p and q can be each cut into a finite number of pieces p pn and q qm with each pi congruent to qi by a rigid euclidean motion provided that p and q have the same volume 18 this question was answered in the negative by max dehn 2 4 29 who found a functionlal on polytopes that is invariant under dissections over rigid motions while varying in value among polytopes of equal volume in other words the dehn invariant is a simple rigid motion invariant valuation on polytopes that is not equal to volume under any normalization dehn s solution left open the question of exactly what conditions oil p and q imply equidissectability over the group of rigid motions although this problem was solved by hadwiger in the case where only translations and no rotations nor reflections are permitted see 2 16 17 26 29 in the course of studying this and related problems hadwiger discovered a characterization of euclidean volume as a continuous rigid motion invariant simple valuation on compact convex sets that is a continuous rigid motion invariant valuation that vanishes on convex sets of less than full dimension this result led in turn to a complete characterization of all continuous rigid motion inivariant valuations on compact convex sets in 1r as consisting of a real n 4 dimensional vector space spanned by the intrinsic volumes or quermassintegrals 16 also 20 21 31 since many standard functionals and integral operators can be interpreted as invariant valuations such as intrinsic volumes mean projections crofton and kinematic formulas what came to be known as hadwiger s chtracterization theorem proved to be a valuable tool for generating quick and effortless proofs of many formulas and equations in integral geometry unfortunately hadwiger s original proof was long and difficult 161 while seeking a shorter proof of hadwiger s volume characterization the author discovered received by the editors june 24 1996 and in revised form september 29 1997 1991 mathematics subject classification primary 52a22 52a38 52a39 52b45 research supported in part by nsf grants dms 9022140 to msri and dms 9626688 to the author 1999 american mathematical society 71 this content downloaded from 207 46 13 52 on mon 24 oct 2016 04 15 57 utc all use subject to http about jstor org terms

valuations on manifolds and rumin cohomology smooth valuations on manifolds are studied by establishing a link with the rumin de rham complex of the co sphere bundle several operations on differential forms induce operations on smooth valuations signature operator rumin laplace operator euler verdier involution and derivation operator as an application alesker s hard lefschetz theorem for even translation invariant valuations on a finite dimensional euclidean space is generalized to all translation invariant valuations the proof uses kahler identities the rumin de rham complex and spectral geometry

affine vs euclidean isoperimetric inequalities abstract it is shown that every even zonal measure on the euclidean unit sphere gives rise to an isoperimetric inequality for sets of finite perimeter which directly implies the classical euclidean isoperimetric inequality the strongest member of this large family of inequalities is shown to be the only affine invariant one among them the petty projection inequality as an application a family of sharp sobolev inequalities for functions of bounded variation is obtained each of which is stronger than the classical sobolev inequality moreover corresponding families of l p isoperimetric and sobolev type inequalities are also established

projection bodies in complex vector spaces the space of minkowski valuations on an m dimensional complex vector space which are continuous translation invariant and contravariant under the complex special linear group is explicitly described each valuation with these properties is shown to satisfy geometric inequalities of the brunn minkowski aleksandrov fenchel and minkowski type

lutwak petty projection inequalities for minkowski valuations and their duals abstract lutwak s volume inequalities for polar projection bodies of all orders are generalized to polarizations of minkowski valuations generated by even zonal measures on the euclidean unit sphere this is based on analogues of mixed projection bodies for such minkowski valuations and a generalization of the notion of centroid bodies a new integral representation is used to single out lutwak s inequalities as the strongest among these families of inequalities which in turn are related to a conjecture on affine quermassintegrals in the dual setting a generalization of volume inequalities for intersection bodies of all orders by leng and lu is proved these results are related to grinberg s inequalities for dual affine quermassintegrals

lp minkowski valuations on polytopes abstract for 1 p ludwig haberl and parapatits classified l p minkowski valuations intertwining the special linear group with additional conditions such as homogeneity and continuity in this paper a complete classification of l p minkowski valuations intertwining the special linear group on polytopes without any additional conditions is established for p 1 including p for n 3 and p 1 there exist valuations not mentioned before

contact measures in isotropic spaces abstract we revisit the contact measures introduced by firey and further developed by schneider and teufel from the perspective of the theory of valuations on manifolds this reveals a link between the kinematic formulas for area measures studied by wannerer and the integral geometry of curved isotropic spaces as an application we find explicitly the kinematic formula for the surface area measure in hermitian space

projection bodies and valuations let be the projection operator which maps every polytope to its projection body it is well known that maps the set of polytopes pn in rn into pn that it is a valuation and that for every p pn p is affinely associated to p it is shown that these properties characterize the projection operator this proves a conjecture by lutwak

orlicz projection bodies abstract minkowski s projection bodies have evolved into l p projection bodies and their asymmetric analogs these all turn out to be part of a far larger class of orlicz projection bodies the analog of the classical petty projection inequality is established for the new orlicz projection bodies

hermitian integral geometry we give in explicit form the principal kinematic formula for the action of the ane unitary group on c n together with a straightforward algebraic method for computing the full array of unitary kinematic formulas expressed in terms of certain convex valuations introduced essentially by h tasaki we introduce also several other canonical bases for the algebra of unitary invariant valuations explore their interrelations and characterize in these terms the cones of positive and monotone elements

moments and valuations all measurable and rm sl n covariant vector valued valuations on convex polytopes containing the origin in their interiors are completely classified the moment vector is shown to be essentially the only such valuation

a valuation theoretic approach to translative equidecomposability abstract all simple translation invariant valuations on polytopes are classified as a direct consequence the well known conditions for translative equidecomposability are recovered furthermore a simplified proof of the classification of continuous simple translation invariant valuations is presented

competition and efficiency in congested markets we study the efficiency of oligopoly equilibria oe in congested markets the motivating examples are the allocation of network flows in a communication network or of traffic in a transportation network we show that increasing competition among oligopolists can reduce efficiency measured as the difference between users willingness to pay and delay costs we characterize a tight bound of 5 6 on efficiency in pure strategy equilibria when there is zero latency at zero flow and a tight bound of 2 2 2 with positive latency at zero flow these bounds are tight even when the numbers of routes and oligopolists are arbitrarily large

lagrangean variables in infinite dimensional spaces for a dynamic economic equilibrium problem abstract this paper is focused on the study of a dynamic competitive equilibrium by using lagrangean multipliers this mathematical formulation allows us the improve the walrasian model by considering the common possibility of an uncharged delayed payment in a given time for example by using a credit card firstly the economic equilibrium problem is reformulated as an evolutionary variational problem then the lagrangean theory in infinite dimensional spaces is applied thanks to the application of this theory we obtain the existence of lagrangean multipliers which allows us to give a computational procedure for the equilibrium solutions

general l_p affine isoperimetric inequalities sharp l p affine isoperimetric inequalities are established for the entire class of l p projection bodies and the entire class of l p centroid bodies these new inequalities strengthen the l p petty projection and the l p busemann petty centroid inequality

classification of invariant valuations on the quaternionic plane abstract we describe the orbit space of the action of the group sp 2 sp 1 on the real grassmann manifolds gr k h 2 in terms of certain quaternionic matrices of moore rank not larger than 2 we then give a complete classification of valuations on the quaternionic plane h 2 which are invariant under the action of the group sp 2 sp 1

lp affine isoperimetric inequalities the lp analogues of the petty projection inequality and the busemannpetty centroid inequality are established an affine isoperimetric inequality compares two functionals associated with convex or more general bodies where the ratio of the functionals is invariant under non degenerate linear transformations these isoperimetric inequalities are more powerful than their better known euclidean relatives

efficiency loss in a network resource allocation game we explore the properties of a congestion game in which users of a congested resource anticipate the effect of their actions on the price of the resource when users are sharing a single resource we establish that the aggregate utility received by the users is at least 3 4 of the maximum possible aggregate utility we also consider extensions to a network context where users submit individual payments for each link in the network they may wish to use in this network model we again show that the selfish behavior of the users leads to an aggregate utility that is no worse than 3 4 of the maximum possible aggregate utility we also show that the same analysis extends to a wide class of resource allocation systems where end users simultaneously require multiple scarce resources these results form part of a growing literature on the price of anarchy i e the extent to which selfish behavior affects system efficiency