structural properties of minimax controllers for a class of differential games arising in nonlinear h control abstract this paper introduces in precise mathematical terms two properties named certainty equivalence and generalized certainty equivalence that nonlinear minimax controller problems might possess the certainty equivalence is a generalization of the one introduced earlier by basar and bernhard 1991 and bernhard 1990 which applies to problems where the worst case disturbance may not be unique but the worst case state trajectory is the generalized certainty equivalance on the other hand extends this to accomodate nonunique worst case state trajectories and leads to the constrollers that guarantee a bounded upper value for the underlying game the paper also shows that for a large class of games and under certain conditions certainty equivalent as well as generalized certainty equivalent controllers admit infinite dimensional estimator kalman filter structures where the estimator gain depends on the state of the estimator

estimates for measures of lower dimensional sections of convex bodies abstract we present an alternative approach to some results of koldobsky on measures of sections of symmetric convex bodies which allows us to extend them to the not necessarily symmetric setting we prove that if k is a convex body in r n with 0 int k and is a measure on r n with a locally integrable non negative density g on r n then k c n k k max f g n n k k f k k n for every 1 k n 1 also if is even and log concave and if k is a symmetric convex body in r n and d is a compact subset of r n such that k f d f for all f g n n k then k c k l n k k d where l s is the maximal isotropic constant of a convex body in r s our method employs a generalized blaschke petkantschin formula and estimates for the dual affine quermassintegrals

subelliptic p harmonic maps into spheres and the ghost of hardy spaces the story begins with the paper of muller 59 who for the sake of an application to nonlinear elasticity proved that if the jacobian determinant ju of a sobolev map u w 1 n loc rn rn is nonnegative then it belongs locally to l log l the result is quite intriguing since a priori holder inequality implies only that ju l1 and one does not suspect any higher integrability if one does not assume that ju is nonnegative then as coifman lions meyer and semmes 14 have proved ju belongs to the local hardy space h 1 loc since a nonnegative function belongs to the local hardy space if and only if it belongs locally to l log l the result of coifman lions meyer and semmes generalizes that of muller in fact coifman lions meyer and semmes proved more namely for 1 p the scalar product of a divergence free vector field e lp rn rn and a curl free vector field b lp p 1 rn rn belongs to the hardy space h 1 a priori holder inequality implies integrability of this expression only note that the jacobian is of this form the discovery that many algebraic expressions involving partial derivatives belong to the hardy space turned out to be important and widely applicable in nonlinear partial differential equations this was first shown by helein 40 41 42 let us describe briefly his result consider maps u bn s m from the n dimensional ball to the mdimensional sphere such that the p energy of u given by the functional

espaces de lipschitz et inegalites de poincare resume in the setting of infinite graphs and non compact riemannian manifolds we show that suitable families of poincare inequalities yield global embeddings of sobolev spaces into lipschitz spaces as well as trudinger type inequalities this applies for example to cocompact coverings and to manifolds that are roughly isometric to a manifold with nonnegative ricci curvature in the process we give several reformulations of the sobolev inequalities and in particular show their equivalence with some l p faber krahn inequalities we also give an interpretation of some of our results in terms of distances on graphs associated with the l p norm of the gradient

on the essential spectrum of schrodinger operators on riemannian manifolds the main result of this paper is a lower bound for the essential spectrum of schrodinger operators v on riemannian manifolds in particular we obtain conditions on v which imply the discreteness of the spectrum or equivalently the compactness of the resolvent

optimal control of uncertain systems with incomplete information for the disturbances we investigate the problem of optimization of a terminal cost function for a system depending on a control and on two disturbances for which a priori set membership is known the disturbances are of different natures one becomes known to the controller at the current time we called it observable while the other remains unknown no state measurements are available the problem can be viewed as a differential game of min max type where the controller aims at minimization of the objective function by a strategy which depends only on the observable disturbance since the state of the system is not exactly known due to the presence of an unobservable disturbance we reformulate the problem through a set valued dynamics describing the evolution of the current set estimation of the state to reduce the complexity of the problem we pass to a suboptimal problem where the evolution of the state estimation is restricted to a prescribed collection of sets the main result of the paper is a characterization of the value function of this problem through a hamilton jacobi inequality in terms of dini derivatives which implies a convergent scheme for numerical computations as necessary auxiliary tools we provide new results on evolution and viability of tubes in a given collection of sets that may be of independent interest

generalized intersection bodies abstract we study the structures of two types of generalizations of intersection bodies and the problem of whether they are in fact equivalent intersection bodies were introduced by lutwak and played a key role in the solution of the busemann petty problem a natural geometric generalization of this problem considered by zhang led him to introduce one type of generalized intersection bodies a second type was introduced by koldobsky who studied a different analytic generalization of this problem koldobsky also studied the connection between these two types of bodies and noted that an equivalence between these two notions would completely settle the unresolved cases in the generalized busemann petty problem we show that these classes share many identical structural properties proving the same results using integral geometry techniques for zhang s class and fourier transform techniques for koldobsky s class using a functional analytic approach we give several surprising equivalent formulations for the equivalence problem which reveal a deep connection to several fundamental problems in the integral geometry of the grassmann manifold

small ball probability estimates ψ2 behavior and the hyperplane conjecture we introduce a method which leads to upper bounds for the isotropic constant we prove that a positive answer to the hyperplane conjecture is equivalent to some very strong small probability estimates for the euclidean norm on isotropic convex bodies as a consequence of our method we obtain an alternative proof of the result of j bourgain that every 2 body has bounded isotropic constant with a slightly better estimate if k is a symmetric convex body in rn such that q 2 for every sn 1 and every q 2 then lk c log where c 0 is an absolute constant

robust output feedback model predictive control of constrained linear systems this paper provides a solution to the problem of robust output feedback model predictive control of constrained linear discrete time systems in the presence of bounded state and output disturbances the proposed output feedback controller consists of a simple stable luenberger state estimator and a recently developed robustly stabilizing tube based model predictive controller the state estimation error is bounded by an invariant set the tube based controller ensures that all possible realizations of the state trajectory lie in a simple uncertainty tube the center of which is the solution of a nominal disturbance free system and the cross section of which is also invariant satisfaction of the state and input constraints for the original system is guaranteed by employing tighter constraint sets for the nominal system the complexity of the resultant controller is similar to that required for nominal model predictive control

sections of convex bodies the generalized busemann petty problem asks if the volume of i dimensional central section of a centrally symmetric convex body in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is smaller than that of another such body is the volume of the body also smaller it is proved that the answer is negative if 2 i n the case of a 2 dimensional section remains open the proof uses techniques in functional analysis and radon transforms on grassmannians it also requires the notion of an i intersection body which generalizes the notion of an intersection body inequalities among the volumes of projection bodies polar projection bodies and their central sections are proved they are related to the maximal slice problem

gelfand numbers of operators with values in a hilbert space in the paper we prove two inequalities involving gelfand numbers of operators with values in a hilbert space the first inequality is a rademacher version of the main result in pa to 1 which relates the gelfand numbers of an operator from a banach spacex intol 2 n with a certain rademacher average for the dual operator the second inequality states that the gelfand numbers of an operatoru froml 1 n into a hilbert space satisfy the inequality k 1 2 c k u leqq c parallel u parallel log 1 n k 1 2 wherec is a universal constant several applications of these inequalities in the geometry of banach spaces are given

convolutions transforms and convex bodies the paper studies convex bodies and star bodies in bbb r n by using radon transforms on grassmann manifolds p cosine transforms on the unit sphere and convolutions on the rotation group of bbb r n it presents dual mixed volume characterizations of i intersection bodies and l p balls which are related to certain volume inequalities for cross sections of convex bodies it considers approximations of special convex bodies by analytic bodies and various finite sums of ellipsoids which preserve special geometric properties convolution techniques are used to derive formulas for mixed volumes mixed surface measures and p cosine transforms they are also used to prove characterizations of geometric functionals such as surface area and dual quermassintegrals

higher order poincare inequalities associated with linear operators on stratified groups and applications this paper considers the dual of anisotropic sobolev spaces on any stratified groups for 0 k m and every linear bounded functional t on anisotropic sobolev space wm k p on we derive a projection operator l from wm p to the collection k 1 of polynomials of degree less than k 1 such that t xi lu t xiu for all u wm p and multi index i with d i k we then prove a general poincare inequality involving this operator l and the linear functional t as applications we often choose a linear functional t such that the associated l is zero and consequently we can prove poincare inequalities of special interests in particular we obtain poincare inequalities for functions vanishing on tiny sets of positive bessel capacity on stratified groups finally we derive a hedberg wolff type characterization of measures belonging to the dual of the fractional anisotropic sobolev spaces w p

robust stabilization of uncertain systems stabilization of uncertain systems is addressed in this paper we consider a controlled system with bounded dynamics uncertainty the evolution law of which can be written as a differential inclusion in order to stabilize the system we consider a lyapunov candidate function and look for controls within the set of controls satisfying a universal stability condition in order to meet this constraint we differentiate the control law and derive the differential inclusion governing the invariant evolution of the control law inside the set of stabilizing controls we show that under regularity assumptions a set valued evolution law of controls can be found explicitly this enables us to select the minimal norm velocity of control satisfying the universal stability condition and finally to propose a single valued explicit control scheme devoted to robust stabilization of nonlinear systems

intersection bodies and generalized cosine transforms the paper is focused on intimate connection between geometric properties of intersection bodies in convex geometry and generalized cosine transforms in harmonic analysis a new concept of intersection body that unifies some known classes of geometric objects is introduced a parallel between trace theorems in function theory restriction onto lower dimensional subspaces of the spherical radon transforms and the generalized cosine transforms and sections of intersection bodies is established new integral formulas for different classes of cosine transforms are obtained and examples of intersection bodies are given we also revisit some known facts in this area and give them new simple proofs

estimation optimale du gradient du semi groupe de la chaleur sur le groupe de heisenberg abstract en utilisant l inegalite de poincare et la formule de representation on montre que sur le groupe de heisenberg de dimension reelle 3 h 1 il existe une constante c 0 telle que e t f g c e t f g g h 1 t 0 f c o h 1 ce resultat repond par l affirmation a la question ouverte de b k driver t melcher hypoelliptic heat kernel inequalities on the heisenberg group j funct anal 221 2005 340 365 aussi le resultat principal de b k driver t melcher hypoelliptic heat kernel inequalities on the heisenberg group j funct anal 221 2005 340 365 peut etre considere comme une consequence immediate de l inegalite precedente

holomorphic semi groups and the geometry of banach spaces according to a fundamental theorem of dvoretzky cf 24 25 any infinite dimensional banach space x contains for each integer n and each e 0 a subspace xn which is 1 e isomorphic to 12n in the classical banach spaces lp and 1p with 1 p x it is well known that this result can be improved these spaces contain uniformly complemented ln 7 s this means that we can find subspaces xn as above with the additional property that there exist projections p x xn such that supn 11 pn 11 cx moreover it is also well known that this property no longer holds for 11 or lo this motivated the question cf 13 to determine which infinite dimensional banach spaces contain uniformly complemented 1 n s we will prove below that all uniformly convex spaces have this property moreover we will show that a banach space x verifies this property locally see definition 2 10 for more precision if and only if x does not contain l n s uniformly progressively the work of previous authors cf 14 12 and particularly 5 has reduced such so called geometric questions to a purely analytic problem consisting of verifying in x a certain inequality referred to as k convexity in this paper we connect this inequality with the holomorphy of a certain semi group of operators on lp x and we show how this leads to the results already mentioned

slicing inequalities for measures of convex bodies abstract we consider the following problem does there exist an absolute constant c so that for every n n every integer 1 k n every origin symmetric convex body l in r n and every measure with non negative even continuous density in r n 1 l c k max h gr n k l h l k n where gr n k is the grassmanian of n k dimensional subspaces of r n and l stands for volume this question is an extension to arbitrary measures in place of volume and to sections of arbitrary codimension k of the slicing problem a major open problem in convex geometry it was proved in 18 19 that 1 holds for arbitrary origin symmetric convex bodies all k and all with c o n in this article we prove inequality 1 with an absolute constant c for unconditional convex bodies and for duals of bodies with bounded volume ratio we also prove that for every 0 1 there exists a constant c c so that inequality 1 holds for every n n every origin symmetric convex body l in r n every measure with continuous density and the codimension of sections k n the proofs are based on a stability result for generalized intersection bodies and on estimates of the outer volume ratio distance from an arbitrary convex body to the classes of generalized intersection bodies in the last section we show that for some measures the behavior of minimal sections may be very different from the case of volume

harnack estimates for degenerate parabolic equations modeled on the subelliptic p laplacian abstract we establish a harnack inequality for a class of quasi linear pde modeled on the prototype t u i 1 m x i x u p 2 x i u where p 2 x x 1 x m is a system of lipschitz vector fields defined on a smooth manifold m endowed with a borel measure and x i denotes the adjoint of x i with respect to our estimates are derived assuming that i the control distance d generated by x induces the same topology on m ii a doubling condition for the measure of d metric balls and iii the validity of a poincare inequality involving x and our results extend the recent work in 16 36 to a more general setting including the model cases of 1 metrics generated by hormander vector fields and lebesgue measure 2 riemannian manifolds with non negative ricci curvature and riemannian volume forms and 3 metrics generated by non smooth baouendi grushin type vector fields and lebesgue measure in all cases the harnack inequality continues to hold when the lebesgue measure is substituted by any smooth volume form or by measures with densities corresponding to muckenhoupt type weights

isoperimetric and sobolev inequalities for carnot caratheodory spaces and the existence of minimal surfaces after hormander s fundamental paper on hypoellipticity 54 the study of partial differential equations arising from families of noncommuting vector fields has developed significantly in this paper we study some basic functional and geometric properties of general families of vector fields that include the hormander type as a special case similar to their classical counterparts such properties play an important role in the analysis of the relevant differential operators both linear and nonlinear to motivate our results we recall some classical inequalities let e c r be a caccioppoli set a measurable set having a locally finite perimeter then one has the isoperimetric inequality

monotonicity theorems for laplace beltrami operator on riemannian manifolds abstract for free boundary problems on euclidean spaces the monotonicity formulas of alt caffarelli friedman and caffarelli jerison kenig are cornerstones for the regularity theory as well as the existence theory in this article we establish the analogs of these results for the laplace beltrami operator on riemannian manifolds as an application we show that our monotonicity theorems can be employed to prove the lipschitz continuity for the solutions of a general class of two phase free boundary problems on riemannian manifolds

an example related to the slicing inequality for general measures abstract for n n let s n be the smallest number s 0 satisfying the inequality k f s k 1 n max s n 1 k f for all centrally symmetric convex bodies k in r n and all even continuous probability densities f on k here k is the volume of k it was proved in 16 that s n 2 n and in analogy with bourgain s slicing problem it was asked whether s n is bounded from above by a universal constant in this note we construct an example showing that s n c n log log n where c 0 is an absolute constant additionally for any 0 2 we describe a related example that satisfies the so called condition

isomorphic properties of intersection bodies we study isomorphic properties of two generalizations of intersection bodies the class ikn of k intersection bodies in rn and the class bpkn of generalized k intersection bodies in rn in particular we show that all convex bodies can be in a certain sense approximated by intersection bodies namely if k is any symmetric convex body in rn and 1 k n 1 then the outer volume ratio distance from k to the class bpkn can be estimated by o v r k bpkn inf c k 1n c bpkn k c cnklogenk where c 0 is an absolute constant next we prove that if k is a symmetric convex body in rn 1 k n 1 and its k intersection body ik k exists and is convex then dbm ik k b2n c k where c k is a constant depending only on k dbm is the banach mazur distance and b2n is the unit euclidean ball in rn this generalizes a well known result of hensley and borell we conclude the paper with volumetric estimates for k intersection bodies

on gradient bounds for the heat kernel on the heisenberg group it is known that the couple formed by the two dimensional brownian motion and its levy area leads to the heat kernel on the heisenberg group which is one of the simplest sub riemannian space the associated diffusion operator is hypoelliptic but not elliptic which makes difficult the derivation of functional inequalities for the heat kernel however driver and melcher and more recently h q li have obtained useful gradient bounds for the heat kernel on the heisenberg group we provide in this paper simple proofs of these bounds and explore their consequences in terms of functional inequalities including cheeger and bobkov type isoperimetric inequalities for the heat kernel

generalized intersection bodies are not equivalent abstract in a koldobsky a functional analytic approach to intersection bodies geom funct anal 10 2000 1507 1526 a koldobsky asked whether two types of generalizations of the notion of an intersection body are in fact equivalent the structures of these two types of generalized intersection bodies have been studied by the author in e milman generalized intersection bodies j funct anal 240 2 2006 530 567 providing substantial evidence for a positive answer to this question the purpose of this note is to construct a counter example which provides a surprising negative answer to this question in a strong sense this implies the existence of non trivial non negative functions in the range of the spherical radon transform and the existence of non trivial spaces which embed in l p for certain negative values of p

subelliptic mollifiers and a basic pointwise estimate of poincare type the aim of this paper is to prove a pointwise estimate which plays a fundamental role in the analysis of both linear and nonlinear partial differential equations arising from a system of non commuting vector fields x x1 xm although our result holds in a larger setting for the unity of presentation we confine ourselves to the specific case of smooth vector fields x1 xm satisfying hormander s finite rank condition h rank lie x1 xm x n for every x n let d n n be the carnot caratheodory metric associated to x1 xm and for x0 n and r 0 set b bd x0 r y n d x0 y 0 the symbol ab will denote the concentric ball bd x0 ar for a given function u we let xu x1u xmu and xu 2 m j 1 xj u 2 throughout the paper we will use the standard notation ub 1 b b u y dy our main result is theorem 1 1 let u n and x0 u there exist c r0 0 and a 1 depending only on u and the vector fields x1 xm such that for any u c 1 ab r r0 and x b one has u x ub c

lq l holder continuity for quasilinear parabolic equations associated to sobolev derivations abstract let x be a locally compact metrizable space endowed with a couple of equivalent finite radon measures m and and let e be a hilbert c monomodule over c0 x we consider a class of abstract nonlinear parabolic equations defined as follows let be a closed derivation from l2 x m to l2 e and tt be the strongly continuous nonlinear semigroup naturally associated in the sense of brezis 1973 to the convex l s c functional e u x u x p d x where is the natural modulus function associated to e the generator of the semigroup considered is a natural generalization of the usual p laplacian operator we suppose that a suitable sobolev like inequality of the form u l2d d 2 x m c u l2 e holds true for some d 2 with p 2 d then tt is a nonlinear markov semigroup in the sense that it is order preserving and nonexpansive on each lq x m for any q 2 and moreover it satisfies ttu ttv l x m cm x t u v lq x m for all q 2 and suitable constants depending only on p q d examples include the semigroup generated by the p laplacian on finite measure manifolds with boundary and homogeneous dirichlet boundary conditions as well as by p laplacian like operators associated both to regular sub riemannian structures and to systems of possibly singular or degenerate vector fields satisfying the appropriate sobolev inequalities

partially observed differential games infinite dimensional hamilton jacobi isaacs equations and nonlinear h_ infty control this paper presents new results for partially observed nonlinear differential games using the concept of information state we solve this problem in terms of an infinite dimensional partial differential equation which turns out to be the hamilton jacobi isaacs hji equation for partially observed differential games we give definitions of smooth and viscosity solutions and prove that the value function is a viscosity solution of the hji equation we prove a verification theorem which implies that the optimal controls are separated in that they depend on the observations through the information state this constitutes a separation principle for partially observed differential games we also present some new results concerning the certainty equivalence principle under certain standard assumptions our results are applied to a nonlinear output feedback h infty robust control problem

gagliardo nirenberg inequalities for differential forms in heisenberg groups the l 1 sobolev inequality states that for compactly supported functions u on the euclidean n space the l n n 1 norm of a compactly supported function is controlled by the l 1 norm of its gradient the generalization to differential forms due to lanzani and stein and bourgain and brezis is recent and states that a the l n n 1 norm of a compactly supported differential h form is controlled by the l 1 norm of its exterior differential du and its exterior codifferential delta u in special cases the l 1 norm must be replaced by the mathcal h 1 hardy norm we shall extend this result to heisenberg groups in the framework of an appropriate complex of differential forms

elliptic differential operators on lie groups abstract each strongly elliptic operator in the enveloping lie algebra associated with a continuous representation of a lie group generates a holomorphic semigroup s whose action is determined by a universal integral kernel k we derive pointwise bounds and lp bounds on k and its derivatives for example if the operator is mth order and the group has dimension d then we prove that k t g at d m e b g m t l m 1 e t for all g g and t 0 where a b and are positive constants the derivative bounds have similar decrease properties as a function over g the correct singularity as a function of t and also reflect the joint analyticity of k one new feature in the analysis is a generalized nash inequality for the differential structure of the group representation and one application is to sobolev embeddings for lie groups

a subelliptic analogue of aronson serrin s harnack inequality we study the harnack inequality for weak solutions of a class of degenerate parabolic quasilinear pde begin aligned partial t u x i a i x t u xu b x t u xu end aligned in cylinders omega times 0 t where omega subset m is an open subset of a manifold m endowed with control metric d corresponding to a system of lipschitz continuous vector fields x x 1 ldots x m and a measure d sigma we show that the harnack inequality follows from the basic hypothesis of doubling condition and a weak poincare inequality in the metric measure space m d d sigma we also show that such hypothesis hold for a class of riemannian metrics g epsilon collapsing to a sub riemannian metric lim epsilon rightarrow 0 g epsilon g 0 uniformly in the parameter epsilon ge 0

small ball probability estimates for log concave measures we establish a small ball probability inequality for isotropic log concave probability measures there exist absolute constants c1 c2 0 such that if x is an isotropic log concave random vector in r with 2 constant bounded by b and if a is a non zero n n matrix then for every e 0 c1 and y r p ax y 2 6 e a hs 6 e c2 b a hs a op 2 where c1 c2 0 are absolute constants

self improving properties of john nirenberg and poincare inequalities on spaces of homogeneous type abstract we give a condition which ensures that if one inequality of sobolev poincare type is valid then other stronger inequalities of a similar type also hold including weighted versions our main result includes many previously known results as special cases we carry out the analysis in the context of spaces of homogeneous type but the main result is new even in the usual euclidean setting

an alternative approach to the holder continuity of solutions to some elliptic equations abstract we give an alternative and self contained proof of the holder continuity of locally bounded solutions to some elliptic equations including the equation associated with the p laplacian operator we combine the techniques introduced by de giorgi and moser with a 1 dimensional poincare inequality

analysis on lie groups theorie locale diffusion de la chaleur sur des groupes de lie connexes la theorie globale les inegalites de sobolev bases graduees sur des corps lisses construction des bases graduees estimation locale du noyau diffusion de la chaleur demonstration de l inegalite de sobolev locale formes de dirichlet sur des groupes le semi groupe de convolution les conditions de sobolev modifiees

surgery of the faber krahn inequality and applications to heat kernel bounds 2 cutting faber krahn inequalities 3 2 1 fk functions 3 2 2 cutting a manifold 6 2 3 non parabolic case 7 2 4 fk functions in balls 9 2 5 relative fk function 16

a n estimate for measures of hyperplane sections of convex bodies the hyperplane or slicing problem asks whether there exists an absolute constant c so that for any origin symmetric convex body k in rn k n 1n cmax sn 1 k where is the central hyperplane in rn perpendicular to and k stands for volume of proper dimension the problem is still open with the best to date estimate c n1 4 established by klartag who slightly improved the previous estimate of bourgain it is much easier to get a weaker estimate with c n in this note we show that the n estimate holds for arbitrary measure in place of volume namely if l is an origin symmetric convex body in rn and is a measure with non negative even continuous density on l then l nnn 1cnmax sn 1 l l 1 n where cn b2n n 1n b2n 1 1 and b2n is the unit euclidean ball in rn we deduce this inequality from a stability result for intersection bodies

global bounds of schrodinger heat kernels with negative potentials abstract we obtain global in time bounds for the heat kernel g of the schrodinger operator l v the potential v satisfies v x c d x b near infinity with b 0 the result can be described as follows suppose l is positive and b 2 then g g x t y 0 has a global upper bound which is a standard gaussian times a polynomial function of x y t however when l is not nonnegative and negative eigenvalues exist g g x t 0 0 is bounded below by a standard gaussian times ectwithc 0 in other words the growth rate of g x t 0 0 is comparable with the heat kernel of c regardless how fast v decays near infinity

a cr poincare inequality on the complex sphere abstract the main purpose of this paper is to prove a cr poincare inequality with sharp exponent on the sphere in complex space we use the complex tangential gradient on the sphere instead of the usual laplace beltrami gradient on the sphere

robust control of set valued discrete time dynamical systems this paper presents results obtained for the control of set valued discrete time dynamical systems such systems model nonlinear systems subject to persistent bounded noise a robust control problem for such systems is introduced the problem is formulated as a dynamic game wherein the controller plays against the set valued system both necessary and sufficient conditions in terms of stationary dynamic programming equalities are presented the output feedback problem is solved using the concept of an information state where a decoupling between estimation and control is obtained the methods yield a conceptual approach for constructing controlled invariant sets and stabilizing controllers for uncertain nonlinear systems

intersection bodies and lp spaces abstract we prove that the convex intersection bodies are isomorphically equivalent to unit balls of subspaces of l q for each q 0 1 this is done by extending to negative values of p the factorization theorem of maurey and nikishin which states that for any 0 p q 1 every banach subspace of l p is isomorphic to a subspace of l q

norm estimates for k plane transforms and geometric inequalities abstract the article is devoted to remarkable interrelation between the norm estimates for k plane transforms in weighted and unweighted l p spaces and geometric integral inequalities for cross sections of measurable sets in r n a similar interrelation is studied for more general j plane to k plane transforms on affine grassmannians and their compact modifications the article contains a series of new integral geometric inequalities with sharp constants explicit equalities conjectures and open problems

the geometry of p convex intersection bodies abstract busemann s theorem states that the intersection body of an origin symmetric convex body is also convex in this paper we provide a version of busemann s theorem for p convex bodies we show that the intersection body of a p convex body is q convex for certain q furthermore we discuss the sharpness of the previous result by constructing an appropriate example this example is also used to show that ik the intersection body of k can be much farther away from the euclidean ball than k finally we extend these theorems to some general measure spaces with log concave and s concave measures

intersection bodies and dual mixed volumes the setting for this article is euclidean n space ky n 2 let x denote the set of convex bodies compact convex subsets with nonempty interiors in r and let x denote the subset of x that contains the centered centrally symmetric with respect to the origin bodies for ue s let e denote the hyperplane through the origin that is orthogonal to u and let ki e denote the image of the orthogonal projection of the body ke x onto e we shall use v to denote i dimensional lebesgue measure for v and v we shall usually write v and u shephard 38 poses the question if k l e xe and

generalizations of the busemann petty problem for sections of convex bodies abstract we present generalizations of the busemann petty problem for dual volumes of intermediate central sections of symmetric convex bodies it is proved that the answer is negative when the dimension of the sections is greater than or equal to 4 for two three dimensional sections both negative and positive answers are given depending on the orders of dual volumes involved and certain cases remain open for bodies of revolution a complete solution is obtained in all dimensions

operateurs uniformement sous elliptiques sur les groupes de lie given a connected unimodular lie group g having polynomial volume growth and left invariant vector fields x1 xk which generate the lie algebra of g consider the operator la i j 1kxiaijxj where a g rk rk is a smooth symmetric matrix valued function which satisfies a 2 i j 1k aij x i j 1 2 x g rk for some 0 1 what we show is that the corresponding heat flow semigroup etla admits a kernel which satisfies two sided gaussian estimates in terms of the control distance determined by the xi s moreover the estimates can be made to depend on a only through we also prove harnack inequalities and holder regularity of solutions we end with a discussion of some other sub elliptic situations

intersection bodies positive definite distributions and the busemann petty problem the 1956 busemann petty problem asks whether symmetric convex bodies with larger central hyperplane sections also have greater volume in 1988 lutwak introduced the concept of an intersection body which is closely related to the busemann petty problem we prove that an origin symmetric star body k in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is an intersection body if and only if x 1 k is a positive definite distribution on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i where x k min a 0 x ak we use this result to show that for every dimension n there exist polytopes in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i which are intersection bodies for example the cross polytope the unit ball of every subspace of l p 0 p 2 is an intersection body the unit ball of the space l n q 2 q n 5 using lutwak s connection with the busemann petty problem we present new counterexamples to the problem for n 5 and confirm the conjecture of meyer that the answer to the problem is affirmative if the smaller body is a polar projection body

spectral convergence under bounded ricci curvature abstract for a noncollapsed gromov hausdorff convergent sequence of riemannian manifolds with a uniform bound of ricci curvature we establish two spectral convergence one of them is on the hodge laplacian acting on differential one forms the other is on the connection laplacian acting on tensor fields of every type which include all differential forms these are sharp generalizations of cheeger colding s spectral convergence of the laplacian acting on functions to the cases of tensor fields and differential forms these spectral convergence have two direct corollaries one of them is to give new bounds on such eigenvalues in terms of bounds on volume diameter and the ricci curvature the other is that we show the upper semicontinuity of the first betti numbers with respect to the gromov hausdorff topology and give the equivalence between the continuity of them and the existence of a uniform spectral gap on the other hand we also define measurable curvature tensors of the noncollapsed gromov hausdorff limit space of a sequence of riemannian manifolds with a uniform bound of ricci curvature which include riemannian curvature tensor the ricci curvature and the scalar curvature as fundamental properties of our ricci curvature we show that the ricci curvature coincides with the difference between the hodge laplacian and the connection laplacian and is compatible with gigli s one and lott s ricci measure moreover we prove a lower bound of the ricci curvature is compatible with a reduced riemannian curvature dimension condition we also give a positive answer to lott s question on the behavior of the scalar curvature with respect to the gromov hausdorff topology by using our scalar curvature

the dual brunn minkowski theory for bounded borel sets dual affine quermassintegrals and inequalities abstract this paper develops a significant extension of e lutwak s dual brunn minkowski theory originally applicable only to star shaped sets to the class of bounded borel sets the focus is on expressions and inequalities involving chord power integrals random simplex integrals and dual affine quermassintegrals new inequalities obtained include those of isoperimetric and brunn minkowski type a new generalization of the well known busemann intersection inequality is also proved particular attention is given to precise equality conditions which require results stating that a bounded borel set almost all of whose sections of a fixed dimension are essentially convex is itself essentially convex

sharp a priori estimates for div curl systems in heisenberg groups abstract in this paper we prove a family of inequalities for differential forms in heisenberg groups h 1 and h 2 that are the natural counterpart of a class of div curl inequalities in de rham s complex proved by lanzani stein and bourgain brezis

sub elliptic global high order poincare inequalities in stratified lie groups and applications sharp poincare inequalities on balls or chain type bounded domains have been extensively studied both in classical euclidean space and carnot caratheodory spaces associated with sub elliptic vector fields e g vector fields satisfying hormander s condition in this paper we investigate the validity of sharp global poincare inequalities of both first order and higher order on the entire nilpotent stratified lie groups or on unbounded extension domains in such groups we will show that simultaneous sharp global poincare inequalities also hold and weighted versions of such results remain to be true more precisely let g be a nilpotent stratified lie group and f be in the localized non isotropic sobolev space w m p loc g w here 1 p q mand q is the homogeneous dimension of the lie group g suppose that the mth sub elliptic derivatives of f is globally l p integrable i e g x m f x p dx is finite but assume that lower order sub elliptic derivatives are only locally l p integrable we denote the space of such functions as b m p g we prove a high order poincare inequality for f minus a polynomial of order m 1 over the entire space g or unbounded extension domains as applications we will prove a density theorem stating that smooth functions with compact support are dense in b m p g modulus a finite dimensional subspace

quasiconformal maps in metric spaces with controlled geometry this paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry the principal message is that such a theory is both relevant and viable the first main issue is the problem of definition which we next describe quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount this requirement makes sense in every metric space given a homeomorphism f from a metric space x to a metric space y then for x x and r 0 set

robust h sub spl infin output feedback control for nonlinear systems in this paper we present a new approach to the solution of the output feedback robust h sub spl infin control problem we employ the recently developed concept of information state for output feedback dynamic games and obtain necessary and sufficient conditions for the solution to the robust control problem expressed in terms of the information state the resulting controller is an information state feedback controller and is intrinsically infinite dimensional stability results are obtained using the theory of dissipative systems and our results are expressed in terms of dissipation inequalities

a probabilistic take on isoperimetric type inequalities abstract we extend a theorem of groemer on the expected volume of a random polytope in a convex body the extension involves various ways of generating random convex sets we also treat the case of absolutely continuous probability measures rather than convex bodies as an application we obtain a new proof of a recent result due to lutwak yang and zhang on the volume of orlicz centroid bodies

the busemann petty problem for arbitrary measures the busemann petty problem asks whether symmetric convex bodies in n with smaller n 1 dimensional volume of central hyperplane sections necessarily have smaller n dimensional volume the answer to this problem is affirmative for n 4 and negative for n 5 in this paper we generalize the busemann petty problem to essentially arbitrary measure in place of the volume we also present applications of the latter result by proving several inequalities concerning the measure of sections of convex symmetric bodies in n

neighborhoods on the grassmannian of marginals with bounded isotropic constant abstract we show that for any isotropic log concave probability measure on r n for every e 0 every 1 k n and any e g n k there exists f g n k with d e f e and l f c e

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

optimal control of discrete time uncertain systems with imperfect measurement we consider a discrete time optimal control problem in the presence of uncertainties in the dynamics and in the measurement an optimal in min max sense output feedback controller is derived from a discrete bellman equation we propose a formulation of the last equation in the estimation space which allows approximation by a reduced order equation corresponding to a prescribed collection of estimation sets the approach is illustrated up to numerical experiments by a pursuit evasion game on the plane with incomplete information of the current state of the evader

centroid bodies and the logarithmic laplace transform a unified approach abstract we unify and slightly improve several bounds on the isotropic constant of high dimensional convex bodies in particular a linear dependence on the body s 2 constant is obtained along the way we present some new bounds on the volume of l p centroid bodies and yet another equivalent formulation of bourgain s hyperplane conjecture our method is a combination of the l p centroid body technique of paouris and the logarithmic laplace transform technique of the first named author

linear parabolic equations with strong singular potentials using an extension of a recent method of cabre and martel 1999 we extend the blow up and existence result in the paper of baras and goldstein 1984 to parabolic equations with variable leading coefficients under almost optimal conditions on the singular potentials this problem has been left open in baras and goldstein these potentials lie at a borderline case where standard theories such as the strong maximum principle and boundedness of weak solutions fail even in the special case when the leading operator is the laplacian we extend a recent result in cabre and martel from bounded smooth domains to unbounded nonsmooth domains