sufficiency and duality in nonsmooth multiobjective optimization involving generalized f α π d type i functions in this paper new classes of generalized f a p d type i functions are introduced for a nonsmooth multiobjective programming problem based upon these generalized functions sufficient optimality conditions are established weak strong and strict converse duality theorems are also derived for wolfe and mond weir type multiobjective dual programs in order to relate the efficient and weak efficient solutions of primal and dual problems

volume inequalities for isotropic measures a direct approach to ball s simplex inequality is presented this approach which does not use the brascamp lieb inequality also gives barthe s characterization of the simplex for ball s inequality and extends it from discrete to arbitrary measures it also yields the dual inequality along with equality conditions and it does both for arbitrary measures

a variational like inequality for multifunctions with applications abstract we study a variational like inequality problem for a multifunction a certain relevant complementarity problem is defined we establish some existence theorems for the above two problems an application to convex mathematical programming is also shown

general affine surface areas abstract two families of general affine surface areas are introduced basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for l affine surface areas are established

symmetric duality for a class of multiobjective fractional programming problems this paper is concerned with symmetric duality for a class of nondifferentiable multiobjective fractional programming problems two weak duality theorems and two strong duality theorems are proved discussion on some special cases shows that results in this paper extend previous work in this area

inequalities for mixed p affine surface area we prove new alexandrov fenchel type inequalities and new affine isoperimetric inequalities for mixed p affine surface areas we introduce a new class of bodies the illumination surface bodies and establish some of their properties we show for instance that they are not necessarily convex we give geometric interpretations of l p affine surface areas mixed p affine surface areas and other functionals via these bodies the surprising new element is that not necessarily convex bodies provide the tool for these interpretations

a classification of sl n invariant valuations a classification of upper semicontinuous and sl n invariant valuations on the space of n dimensional convex bodies is established as a consequence complete characterizations of cen t ro affine and l p affine surface areas are obtained the proofs make use of a new sl n shaping process for convex bodies

new orlicz affine isoperimetric inequalities abstract the orlicz brunn minkowski theory received considerable attention recently and many results in the l p brunn minkowski theory have been extended to their orlicz counterparts the aim of this paper is to develop orlicz l affine and geominimal surface areas for a single convex body as well as for multiple convex bodies which generalize the l p mixed affine and geominimal surface areas fundamental concepts in the l p brunn minkowski theory our extensions are different from the general affine surface areas by ludwig in 21 moreover our definitions for orlicz l affine and geominimal surface areas reveal that these affine invariants are essentially the infimum supremum of v k l the orlicz mixed volume of k and the polar body of l where l runs over all star bodies and all convex bodies respectively with volume of l equal to the volume of the unit euclidean ball b 2 n properties for the orlicz l affine and geominimal surface areas such as affine invariance and monotonicity are proved related orlicz affine isoperimetric inequalities are also established

generalized mixed quasi variational like inequalities in this paper we use the auxiliary principle technique to suggest and analyze some iterative methods for solving generalized mixed quasi variational like inequalities we show that the convergence of the proposed method requires either pseudomonotonicity or partially relaxed strongly monotonicity under some mild condition as special cases one can obtain a number of new and known algorithms for solving variational inequalities and complementarity problems our results represent significant and important refinements of the previously known results

efficiency through variational like inequalities with lipschitz functions in this work first we introduce several notions of invexity and pseudoinvexity for a locally lipschitz function by means of the generalized jacobian we study relationships between these concepts in particular the implications between preinvexity and invexity next we obtain necessary and sufficient optimality conditions for efficient and weak efficient solutions of finite dimensional non necessarily pareto vector optimization problems with locally lipschitz objective functions through solutions of vector variational like inequality problems these conditions are stated via the generalized jacobian and under pseudoinvexity hypotheses and they show that a vector optimization problem can be reformulated as a vector variational like inequality problem this work extends and improves several previous papers where the objective function of the vector optimization problem is assumed to be differentiable or being locally lipschitz the authors consider the componentwise subdifferential based on the clarke s generalized gradients of the components of the objective function throughout the paper some simple examples are given in order to illustrate the main concepts and results

symmetric duality for a class of nondifferentiable multi objective fractional variational problems abstract we introduce a symmetric dual pair for a class of nondifferentiable multi objective fractional variational problems weak strong converse and self duality relations are established under certain invexity assumptions the paper includes extensions of previous symmetric duality results for multi objective fractional variational problems obtained by kim lee and schaible d s kim w j lee s schaible symmetric duality for invex multiobjective fractional variational problems j math anal appl 289 2004 505 521 and symmetric duality results for the static case obtained by yang wang and deng x m yang s y wang x t deng symmetric duality for a class of multiobjective fractional programming problems j math anal appl 274 2002 279 295 to the dynamic case

ellipsoids and matrix valued valuations a classification is obtained of borel measurable gl n covariant symmetric matrix valued valuations on the space of n dimensional convex polytopes the only ones turn out to be the moment matrix corresponding to the classical legendre ellipsoid and the matrix corresponding to the ellipsoid recently discovered by lutwak yang and zhang a classical concept from mechanics is the legendre ellipsoid or ellipsoid of inertia 2k associated with a convex body k r it can be defined as the unique ellipsoid centered at the center of mass of k such that the ellipsoid s moment of inertia about any axis passing through the center of mass is the same as that of k if we fix a scalar product x y for x y r 2k can be defined by the moment matrix m2 k of k this is the n n matrix with coefficients

symmetric duality for multiobjective fractional variational problems with generalized invexity the concept of symmetric duality for multiobjective fractional problems has been extended to the class of multiobjective variational problems weak strong and converse duality theorems are proved under generalized invexity assumptions a close relationship between these problems and multiobjective fractional symmetric dual problems is also presented

hemiequilibrium like problems abstract in this paper we introduce and consider a new class of equilibrium problems known as hemiequilibrium like problems this new class includes hemiequilibrium equilibrium like problems and several classes of variational inequalities as special cases a number of iterative methods for solving hemiequilibrium like problems are suggested and analyzed by using the auxiliary principle technique we also study the convergence analysis of these iterative methods under some mild conditions the results obtained in this paper can be considered as a novel application of the auxiliary principle technique

relationships between pre invex concepts abstract it is well known that some properties of convex optimization problems especially sufficient conditions for optimality and duality conditions are preserved when convexity is weakened to pre invexity however there are several distinct pre invex concepts b vex b pre invex pre univex p r pre invex and these are not equivalent the pre invexity concepts mentioned above have also some properties which are characteristic and invariant for functions belonging to these classes of functions

non differentiable minimax fractional programming with generalized α univexity in this paper we study a non differentiable minimax fractional programming problem under the assumption of generalized a univex function in this paper we extend the concept of a invexity m a noor on generalized preinvex functions and monotonicities j inequalities pure appl math 5 2004 1 9 and pseudo a invexity s k mishra m a noor on vector variational like inequality problems j math anal appl 311 2005 69 75 to a univexity and pseudo a univexity from a view point of generalized convexity we also introduce the concept of strict pseudo a univex and quasi a univex functions we derive karush kuhn tucker type sufficient optimality conditions and establish weak strong and converse duality theorems for the problem and its three different form of dual problems the results in this paper extend a few known results in the literature

on minimax fractional optimality conditions with invexity abstract under different forms of invexity conditions sufficient kuhn tucker conditions and three dual models are presented for the minimax fractional programming

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

hemivariational like inequalities in this paper we introduce and consider a new class of variational inequalities known as the hemivariational like inequalities it is shown that the hemivariational like inequalities include hemivariational inequalities variational like inequalities and the classical variational inequalities as special cases the auxiliary principle is used to suggest and analyze some iterative methods for solving hemivariational like inequalities under mild conditions the results obtained in this paper can be considered as a novel application of the auxiliary principle technique

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

the sharp sobolev and isoperimetric inequalities split twice this paper shows that each of the sharp endpoint sobolev inequality and the isoperimetric inequality can be split into two sharp and stronger inequalities through either the 1 variational capacity or the 1 integral affine surface area furthermore some related sharp analytic and geometric inequalities are also explored

a characterization of pseudoinvexity for the efficiency in non differentiable multiobjective problems duality abstract we prove that in order for the kuhn tucker or fritz john points to be efficient solutions it is necessary and sufficient that the non differentiable multiobjective problem functions belong to new classes of functions that we introduce here kt pseudoinvex ii or fj pseudoinvex ii respectively we illustrate it by examples these characterizations generalize recent results given for the differentiable case we study the dual problem and establish weak strong and converse duality results

on variational problems characterization of solutions and duality in this paper we introduce a new class of pseudoinvex functions for variational problems using this new concept we obtain a necessary and sufficient condition for a critical point of the variational problem to be an optimal solution illustrated with an example also weak strong and converse duality are established

a note on invex problems with nonnegative variable we provide sufficient optimality conditions involving invexity for mathematical programming problems with nonnegative variable we also show that upon certain assumptions on the scaling function these problems are convex transformable hence certain associated convex problems should be solved some applications are shown

inversion of fractional integrals related to the spherical radon transform explicit inversion formulas are obtained for the analytic family of fractional integrals t f x n sn xy 1 f y dyon the unit sphere in rn 1 arbitrary complex andn 2 are considered in the ease 0 the integralt fcoincides with the spherical radon transform for 1 1 3 5 such integrals are known as the blaschke levy representations and arise in convex geometry probability and the banach space theory for 1 3 5 the integralt fis defined by continuity as the spherical convolution with the power logarithmic kernel different inversion methods are discussed

dual orlicz brunn minkowski theory abstract in this paper a dual orlicz brunn minkowski theory is presented an orlicz radial sum and dual orlicz mixed volumes are introduced the dual orlicz minkowski inequality and the dual orlicz brunn minkowski inequality are established the variational formula for the volume with respect to the orlicz radial sum is proved the equivalence between the dual orlicz minkowski inequality and the dual orlicz brunn minkowski inequality is demonstrated orlicz intersection bodies are defined and the orlicz busemann petty problem is posed

a new ellipsoid associated with convex bodies it is shown that corresponding to each convex body there is an ellipsoid that is in a sense dual to the legendre ellipsoid of classical mechanics sharp affine isoperimetric inequalities are obtained between the volume of the convex body and that of its corresponding new ellipsoid these inequalities provide exact bounds for the isotropic constant associated with the new ellipsoid among other things this leads to a new approach to establishing ball s maximal shadows conjecture for symmetric convex bodies corresponding to each origin symmetric convex or more general subset of euclidean n space r there is a unique ellipsoid with the following property the moment of inertia of the ellipsoid and the moment of inertia of the convex set is the same about every 1 dimensional subspace of r this ellipsoid is called the legendre ellipsoid of the convex set the legendre ellipsoid and its polar the binet ellipsoid are well known concepts from classical mechanics see milman and pajor mpa1 mpa2 lindenstrauss and milman lim and leichtweis le for some historical references it has slowly come to be recognized that along side the brunn minkowski theory there is a dual theory the nature of the duality between this dual brunn minkowski theory and the brunn minkowski theory is subtle and not yet understood it is easily seen that the legendre and binet ellipsoid is an object of this dual brunnminkowski theory this observation leads immediately to the natural question regarding the possible existence of a dual analog of the classical legendre ellipsoid in the brunn minkowski theory it is the aim of this paper to demonstrate the existence of precisely this dual object in retrospect one may well wonder why research supported in part by nsf grant dms 9803261 typeset by ams tex 1 2 a new ellipsoid associated with convex bodies the new ellipsoid presented in this note was not discovered long ago the simple answer is that the definition of the new ellipsoid becomes obvious only with the notion of l2 curvature in hand however the brunn minkowski theory was only recently extended to incorporate the new notion of lp curvature see l2 l3 a positive definite n n real symmetric matrix a generates an ellipsoid a in r defined by a x r x ax 1 where x ax denotes the standard inner product of x and ax in r associated with a star shaped about the origin set k r is its legedre ellipsoid 2k which is generated by the matrix mij k where mij k n 2 v k k ei x ej x dx with e1 en denoting the standard basis for r and v k denoting the ndimensional volume of k we will associate a new ellipsoid 2k with each convex body k r one approach to defining 2k without introducing new notation is to first define it for polytopes and then use approximation with respect to the hausdorff metric to extend the definition to all convex bodies suppose p r is a polytope that contains the origin in its interior let u1 un denote the outer unit normals to the faces of p let a1 an denote the areas i e n 1 dimensional volumes of the corresponding faces and let h1 hn denote the distances from the origin to the corresponding faces the ellipsoid 2p is generated by the matrix mij p where mij p 1 v p n l 1 al hl ei ul ej ul an alternate definition of the operator 2 will be given after additional notation is introduced the easily established affine nature of the operator 2 is formally stated in lemma 1 if k r is star shaped about the origin then for each gl n

subword complexity and projection bodies abstract a polytope p 0 1 d and an 0 1 d induce a so called hartman sequence h p 0 1 z which is by definition 1 at the kth position if k mod 1 p and 0 otherwise k z we prove an asymptotic formula for the subword complexity of such a hartman sequence this result establishes a connection between symbolic dynamics and convex geometry if the polytope p is convex then the subword complexity of h p asymptotically equals the volume of the projection body p of p for almost all 0 1 d

symmetrized double quantum stochastic product integrals a theory is developed of product integrals of the form a s b c t d 1 g h ds dt here a b and c d are disjoint finite subintervals of r and g h is a formal power series in the indeterminate h whose constant term is zero and whose coefficients are elements of l l where l is the space of basic differentials of a multidimensional quantum stochastic calculus the product integrals are themselves formal power series in h whose coefficients are finite sums of iterated stochastic integrals against the elements of l they are symmetrized in such a way that a s b c t d 1 g h ds dt is the image obtained by applying the representation j a b j c d to the coefficients where j a b is the representation canonically associated with the interval a b of a formal power series 1 dg h whose coefficients lie in u u where u is the universal enveloping algebra of the lie algebra l it is shown that the naturally conjectured multiplication rule analogous to the multiplication rule for simple prod

existence and iterative approximations of solutions for mixed quasi variational like inequalities in banach spaces abstract in this paper we introduce and investigate a new class of mixed quasi variational like inequalities in reflexive banach spaces by applying a minimax inequality due to ding tan and a lemma due to chang we establish some existence and uniqueness results of solution for the mixed quasi variational like inequality next by using a kkm theorem due to fan and an auxiliary principle technique due to cohen we suggest two iterative algorithms and study the convergence criteria of iterative sequences generated by the iterative algorithms our results extend improve and unify several known results in the literature

a positive answer to the busemann petty problem in three dimensions we prove that in e3 the busemann petty problem concerning cen tral sections of centrally symmetric convex bodies has a positive answer together with other results this settles the problem in each dimension

sufficiency and duality in nondifferentiable multiobjective programming involving generalized type i functions abstract recently hachimi and aghezzaf defined generalized f d type i functions a new class of functions that unifies several concepts of generalized type i functions in this paper the generalized f d type i functions are extended to nondifferentiable functions by utilizing the new concepts we obtain several sufficient optimality conditions and prove mixed type and mond weir type duality results for the nondifferentiable multiobjective programming problem

the log brunn minkowski inequality abstract for origin symmetric convex bodies i e the unit balls of finite dimensional banach spaces it is conjectured that there exist a family of inequalities each of which is stronger than the classical brunn minkowski inequality and a family of inequalities each of which is stronger than the classical minkowski mixed volume inequality it is shown that these two families of inequalities are equivalent in that once either of these inequalities is established the other must follow as a consequence all of the conjectured inequalities are established for plane convex bodies

nonsmooth invex functions and sufficient optimality conditions in this paper the various definitions of nonsmooth invex functions are gathered in a general scheme by means of the concept of k directional derivative characterizations of nonsmooth k invexity are derived as well as results concerning constrained optimization without any assumption of convexity of the k directional derivatives

convexity of lp intersection bodies we extend the classical brunn theorem to symmetric moments of convex bodies and use it to prove the convexity of the lp intersection bodies of centered convex bodies

on sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

moment entropy inequalities it is shown that the product of the renyi entropies of two independent random vectors provides a sharp lower bound for the expected value of the moments of the inner product of the random vectors this new inequality contains important geometry such as extensions of one of the fundamental affine isoperimetric inequalities the blaschke santalo inequality

optimality and duality for generalized fractional programming involving nonsmooth f ρ convex functions using a parametric approach we establish necessary and sufficient conditions and derive duality theorems for a class of nonsmooth generalized minimax fractional programming problems containing f r convex functions

nondifferentiable multiobjective programming under generalized d univexity abstract in this paper we are concerned with a nondifferentiable multiobjective programming problem with inequality constraints we introduce four new classes of generalized convex functions by combining the concepts of weak strictly pseudoinvex strong pseudoinvex weak quasi invex weak pseudoinvex and strong quasi invex functions in aghezzaf and hachimi numer funct anal optim 22 2001 775 d invex functions in antczak europ j oper res 137 2002 28 and univex functions in bector et al univex functions and univex nonlinear programming proc admin sci assoc canada 1992 p 115 by utilizing the new concepts we derive a karush kuhn tucker sufficient optimality condition and establish mond weir type and general mond weir type duality results for the nondifferentiable multiobjective programming problem

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

multi objective infinite programming using the theory of abstract optimization problems in infinite dimensional spaces we provide necessary optimality conditions of first and second order for weakly efficient solutions of the multi objective infinite programming problem sufficient conditions are given under invexity assumptions we generalize the notion of kkt invexity for the multi objective infinite problem and show that this notion is a necessary and sufficient condition for every vector kkt solution to be a weakly efficient solution moreover we develop a theorem of the alternative useful for proving some of our results

technical note on symmetric duality in multiobjective programming some remarks on recent results abstract certain shortcomings are pointed out in some recent works of das and nanda european journal of operational research 97 1997 167 171 and kim et al european journal of operational research 107 1998 686 691 which show that these studies are highly restricted

algebraic theory of product integrals in quantum stochastic calculus motivated by the search for solutions of the quantum yang baxter equation an algebraic theory of quantum stochastic product integrals is developed the product integrators are formal power series in an indeterminate h whose coefficients are elements of the lie algebra l labelling the usual integrators of a many dimensional quantum stochastic calculus the product integrals are also formal power series in h whose coefficients are finite iterated additive stochastic integrals which act on the exponential domain in the fock space of the calculus and which represent elements of the universal enveloping algebra u of l they obey a multiplication rule suggested by the quantum ito product formula and are characterized among all such formal power series by a grouplike property

sharp convex lorentz sobolev inequalities new sharp lorentz sobolev inequalities are obtained by convexifying level sets in lorentz integrals via the lp minkowski problem new lp isocapacitary and isoperimetric inequalities are proved for lipschitz star bodies it is shown that the sharp convex lorentz sobolev inequalities are analytic analogues of isocapacitary and isoperimetric inequalities

generalized proper efficiency and duality for a class of nondifferentiable multiobjective variational problems withv invexity abstract a mond weir type dual for a class of nondifferentiable multiobjective variational problems in which every component of the objective function contains a term involving the square root of a certain positive semidefinite quadratic form is considered and various duality results viz weak strong and converse duality theorems are developed for conditionally properly efficient solutions these results are obtained under v invexity assumptions and its generalizations on objective and constraint functions this work extends many results on variational problems established earlier

double product integrals and enriquez quantization of lie bialgebras i the quasitriangular identities let t l be the space of all tensors over a lie algebra l in which the lie bracket is obtained by taking commutators in an associative algebra we show that t l becomes a hopf algebra when equipped with a noncommutative modification of the shuffle product together with the standard coproduct a definition is given of directed double product integrals as iterated single product integrals driven by formal power series with coefficients in the tensor product of l with an appropriate associative algebra for the hopf algebra t l h of formal power series we show that elements r h of t l t l h satisfying id r h r h 13r h 23 id r h r h 13r h 12 and which are unitalized by the counit in either copy of t l can be characterized as such directed double product integrals 1 d d r h where r h is a formal power series with coefficients in l l and vanishing constant term

multiobjective programming under d invexity abstract in this paper a generalization of convexity is considered in the case of nonlinear multiobjective programming problem where the functions involved are nondifferentiable by considering the concept of pareto optimal solution and substituting d invexity for convexity the fritz john type and karush kuhn tucker type necessary optimality conditions and duality in the sense of mond weir and wolfe for nondifferentiable multiobjective programming are given

optimality and duality in nondifferentiable multiobjective optimization involvingd type i and related functions abstract in this paper new classes of functions namely d type i d quasitype i and d pseudo type i are defined for a multiobjective nondifferentiable programming problem kuhn tucker type necessary and sufficient optimality conditions are obtained for a feasible point to be a weak minimum for this problem two duals are formulated and various duality results are given by using the above defined classes of functions considering the concept of a weak minimum

optimality criteria in nonlinear programming involving nonconvex functions abstract in this paper a new class of functions called convex quasiconvex and pseudoconvex are defined and a number of sufficient optimality criteria are obtained for nonlinear programming problems involving these functions

symmetric duality for invex multiobjective fractional variational problems we introduce a symmetric dual for multiobjective fractional variational problems under certain invexity assumptions we establish weak strong and converse duality theorems as well as self duality relations the paper includes an extension of previous symmetric duality results for the static case obtained by weir to the dynamic case

minimax fractional programming under nonsmooth generalized f ρ θ d univexity abstract in this paper we are concerned with a nondifferentiable minimax fractional problem with inequality constraints we introduce a new class of generalized convex function that is nonsmooth generalized f d univex function in the framework of the new concept we derive kuhn tucker type sufficient optimality conditions and establish weak strong and converse duality theorems for the problem and its three different types of dual problems

relative entropy of cone measures and lp centroid bodies let k be a convex body in r n we introduce a new affine invariant which we call k t hat can be found in three different ways a as a limit of normalized lp affine surface areas b as the relative entropy of the cone measure of k and the cone measure of k c as the limit of the volume difference of k and lp centroid bodies we investigate properties of k and of related new invariant quantities in particular we show new affine isoperimetric inequalities and we show an information inequality for convex bodies

pre invex functions in multiple objective optimization and called such functions convexlike if s is a convex set and if f is a convex function then clearly f is convexlike elster and nehse obtained a saddlepoint optimality condition for convexlike mathematical programs hayashi and komiya 9 also considered convexlike functions and developed a gordan type theorem of the alternative involving convexlike functions and in addition considered lagrangian duality for convexlike programs hanson 7 considered differentiable functions f s r for which there exists an n dimensional vector function q q u such that for all x u e s

kt invex control problem abstract in this paper we introduce a new condition on functions of a control problem for which we define a kt invex control problem we prove that a kt invex control problem is characterized in order that a kuhn tucker point is an optimal solution we generalize optimality results of known mathematical programming problems we illustrate these results with examples

volume of the polar of random sets and shadow systems we obtain optimal inequalities for the volume of the polar of random sets generated for instance by the convex hull of independent random vectors in euclidean space extremizers are given by random vectors uniformly distributed in euclidean balls this provides a random extension of the blaschke santalo inequality which in turn can be derived by the law of large numbers the method involves shadow systems their connection to busemann type inequalities and how they interact with functional rearrangement inequalities