extending yager s orness concept for the owa aggregators to other mean operators abstract in this paper we address the problem of defining the orness closeness to the maximum and andness closeness to the minimum of fuzzy aggregation operators in particular we center our study in fuzzy mean aggregators as defined by dubois prade and yager for this purpose we propose the introduction of the orness distribution function orness average value orness extension measure and orness sequence which we believe are valid concepts to characterize this type of aggregation operators we study their application to several well known mean operators and prove as well that in the case of yager s owa operators the orness average value that we propose coincides with yager s original orness definition we also introduce a method for the construction of mean operators from a predefined orness distribution function and make some considerations about the extension of these concepts to the cases of weighted aggregations or aggregations based on fuzzy measures

spectral lagrangian methods for collisional models of non equilibrium statistical states we propose a new spectral lagrangian based deterministic solver for the non linear boltzmann transport equation bte in d dimensions for variable hard sphere vhs collision kernels with conservative or non conservative binary interactions the method is based on symmetries of the fourier transform of the collision integral where the complexity in its computation is reduced to a separate integral over the unit sphere s d 1 the conservation of moments is enforced by lagrangian constraints the resulting scheme implemented in free space is very versatile and adjusts in a very simple manner to several cases that involve energy dissipation due to local micro reversibility inelastic interactions or elastic models of slowing down process our simulations are benchmarked with available exact self similar solutions exact moment equations and analytical estimates for the homogeneous boltzmann equation both for elastic and inelastic vhs interactions benchmarking of the simulations involves the selection of a time self similar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for maxwell molecules as studied first in bobylev et al a v bobylev c cercignani g toscani proof of an asymptotic property of self similar solutions of the boltzmann equation for granular materials journal of statistical physics 111 2003 403 417 and generalized to a wide range of related models in bobylev et al a v bobylev c cercignani i m gamba on the self similar asymptotics for generalized non linear kinetic maxwell models communication in mathematical physics in press url the method also produces accurate results in the case of inelastic diffusive boltzmann equations for hard spheres inelastic collisions under thermal bath where overpopulated non gaussian exponential tails have been conjectured in computations by stochastic methods t v noije m ernst velocity distributions in homogeneously cooling and heated granular fluids granular matter 1 57 1998 m h ernst r brito scaling solutions of inelastic boltzmann equations with over populated high energy tails journal of statistical physics 109 2002 407 432 s j moon m d shattuck j swift velocity distributions and correlations in homogeneously heated granular media physical review e 64 2001 031303 i m gamba s rjasanow w wagner direct simulation of the uniformly heated granular boltzmann equation mathematical and computer modelling 42 2005 683 700 and rigorously proven in gamba et al i m gamba v panferov c villani on the boltzmann equation for diffusively excited granular media communications in mathematical physics 246 2004 503 541 39 and a v bobylev i m gamba v panferov moment inequalities and high energy tails for boltzmann equations with inelastic interactions journal of statistical physics 116 2004 1651 1682

cone volume measure of general centered convex bodies abstract we show that the cone volume measure of a convex body with centroid at the origin satisfies the subspace concentration condition this extends former results obtained in the discrete as well as in the symmetric case and implies among others a conjectured best possible inequality for the u functional of a convex body

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

generalized means as model of compensative connectives abstract empirical investigations have shown that instead of strong logical and and or connectives there is often some degree of compensation in between in this paper the generalized means are proposed as connective operators for fuzzy set theory which easily allow for modelling the degree of compensation in a natural manner including the arithmetic and geometric means as well as the maximum and minimum operators as special cases numerical examples with empirical data form an illustration of this concept some aspects of applications to decision making are discussed

liouville type results of curvature operators of fuchsian convex surfaces abstract in this paper we deduce some liouville type results of curvature operators of fuchsian convex surfaces in minkowski space

necessary subspace concentration conditions for the even dual minkowski problem abstract we prove tight subspace concentration inequalities for the dual curvature measures c q k of an n dimensional origin symmetric convex body for q n 1 this supplements former results obtained in the range q n

valuations on sobolev spaces all affinely covariant convex body valued valuations on the sobolev space w 1 1 bbb r n are completely classified it is shown that there is a unique such valuation for blaschke addition this valuation turns out to be the operator which associates with each function f in w 1 1 bbb r n the unit ball of its optimal sobolev norm

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

the logarithmic minkowski problem in analogy with the classical minkowski problem necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone volume measure of the unit ball of a finite dimensional banach space

a stability result for mean width of lp centroid bodies we give a different proof of a recent result of klartag b klartag a central limit theorem for convex sets invent math 168 1 2007 91 131 concerning the concentration of the volume of a convex body within a thin euclidean shell and proving a conjecture of anttila ball and perissinaki m anttila k ball i perissinaki the central limit problem for convex bodies trans amer math soc 355 12 2003 4723 4735 it is based on the study of the lp centroid bodies we prove an almost isometric reverse holder inequality for their mean width and a refined form of a stability result

brunn minkowski inequalities in product metric measure spaces abstract given one metric measure space x satisfying a linear brunn minkowski inequality and a second one y satisfying a brunn minkowski inequality with exponent p 1 we prove that the product x y with the standard product distance and measure satisfies a brunn minkowski inequality of order 1 1 p 1 under mild conditions on the measures and the assumption that the distances are strictly intrinsic the same result holds when we consider restricted classes of sets we also prove that a linear brunn minkowski inequality is obtained in x y when y satisfies a prekopa leindler inequality in particular we show that the classical brunn minkowski inequality holds for any pair of weakly unconditional sets in r n i e those containing the projection of every point in the set onto every coordinate subspace when we consider the standard distance and the product measure of n one dimensional real measures with positively decreasing densities this yields an improvement of the class of sets satisfying the gaussian brunn minkowski inequality furthermore associated isoperimetric inequalities as well as recently obtained brunn minkowski s inequalities are derived from our results

the dual minkowski problem for symmetric convex bodies abstract the dual minkowski problem for even data asks what are the necessary and sufficient conditions on a prescribed even measure on the unit sphere for it to be the q th dual curvature measure of an origin symmetric convex body in r n a full solution to this is given when 1 q n the necessary and sufficient conditions turn out to be an explicit measure concentration condition to obtain the results a variational approach is used where the functional is the sum of a dual quermassintegral and an entropy integral the proof requires two crucial estimates the first is an estimate of the entropy integral which is obtained by using a spherical partition the second is a sharp estimate of the dual quermassintegrals for a carefully chosen barrier convex body

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

the discrete planar l0 minkowski problem abstract in the discrete setting the l 0 minkowski problem extends the question posed and answered by the classical minkowski s existence theorem for polytopes in particular the planar extension which we address in this paper concerns the existence of a convex polygonal body which contains the origin whose boundary sides have preassigned orientations and each triangle formed by the origin with two consecutive vertices is of prescribed area

the cramer rao inequality for star bodies associated with each body k in euclidean n space mathbb r sp n is an ellipsoid gamma sb 2k called the legendre ellipsoid of k it can be defined as the unique ellipsoid centered at the body s center of mass such that the ellipsoid s moment of inertia about any axis passing through the center of mass is the same as that of the body in an earlier paper the authors showed that corresponding to each convex body k subset mathbb r sp n is a new ellipsoid gamma sb 2 k that is in some sense dual to the legendre ellipsoid the legendre ellipsoid is an object of the dual brunn minkowski theory while the new ellipsoid gamma sb 2 k is the corresponding object of the brunn minkowski theory the present paper has two aims the first is to show that the domain of gamma sb 2 can be extended to star shaped sets the second is to prove that the following relationship exists between the two ellipsoids if k is a star shaped set then gamma sb 2 k subset gamma sb 2k with equality if and only if k is an ellipsoid centered at the origin this inclusion is the geometric analogue of one of the basic inequalities of information theory the cramer rao inequality

the first variation of the total mass of log concave functions and related inequalities abstract on the class of log concave functions on r n endowed with a suitable algebraic structure we study the first variation of the total mass functional which corresponds to the volume of convex bodies when restricted to the subclass of characteristic functions we prove some integral representation formulae for such a first variation which suggest to define in a natural way the notion of area measure for a log concave function in the same framework we obtain a functional counterpart of minkowski s first inequality for convex bodies as corollaries we derive a functional form of the isoperimetric inequality and a family of logarithmic type sobolev inequalities with respect to log concave probability measures finally we propose a suitable functional version of the classical minkowski s problem for convex bodies and prove some partial results towards its solution

optimization of logistic systems using fuzzy weighted aggregation logistic scheduling problems are often multi criteria optimization problems with many contradictory objectives and constraints which cannot be properly described by conventional cost functions the use of fuzzy decision making may improve the performance of this type of systems since it allows an easier and suitable description of the confluence of the different criteria of the scheduling process this paper introduces the application of fuzzy weighted aggregation to formulate the logistic system optimization problem further this paper also extends the application of this framework to different types of optimization methodologies dispatching rules if it is used as a performance index or meta heuristics such as genetic algorithms ga or ant colony optimization aco if it is used as an objective function simulation results show that the fuzzy combination of criteria improves the scheduling results whatever optimization methodology is used

rogers shephard type inequalities for sections abstract in this paper we address the following question given a measure on r n does there exist a constant c 0 such that for any m dimensional subspace h r n and any convex body k r n the following sectional rogers shephard type inequality holds k k h c sup y r n y k h we show that this inequality is affirmative in the class of measures with radially decreasing densities with the constant c n m n m m we also prove marginal inequalities of the rogers shephard type for 1 s concave when 0 s and logarithmically concave functions

a berry esseen type inequality for convex bodies with an unconditional basis suppose x x1 xn is a random vector distributed uniformly in a convex body k subset mathbb r n we assume the normalization mathbb e x i 2 1 for i 1 n the body k is further required to be invariant under coordinate reflections that is we assume that x1 xn has the same distribution as x1 xn for any choice of signs then we show that mathbb e left x sqrt n right 2 leq c 2 where c 4 is a positive universal constant and is the standard euclidean norm in mathbb r n the estimate is tight up to the value of the constant it leads to a berry esseen type bound in the central limit theorem for unconditional convex bodies

volume inequalities of convex bodies from cosine transforms on grassmann manifolds abstract the l p cosine transform on grassmann manifolds naturally induces finite dimensional banach norms whose unit balls are origin symmetric convex bodies in r n reverse isoperimetric type volume inequalities for these bodies are established which extend results from the sphere to grassmann manifolds

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

valuations and surface area measures we consider valuations defined on polytopes containing the origin which have measures on the sphere as values we show that the classical surface area measure is essentially the only such valuation which is sl n contravariant of degree one moreover for all real p an lp version of the above result is established for gl n contravariant valuations of degree p this provides a characterization of the lp surface area measures from the lp brunn minkowski theory mathematics subject classification 52b45 28a10

cone volume measures of polytopes abstract the cone volume measure of a polytope with centroid at the origin is proved to satisfy the subspace concentration condition as a consequence a conjectured a dozen years ago fundamental sharp affine isoperimetric inequality for the u functional is completely established along with its equality conditions

fuzzy clustering with volume prototypes and adaptive cluster merging two extensions to objective function based fuzzy clustering are proposed first the point prototypes are extended to hypervolumes whose size can be fixed or can be determined automatically from the data being clustered it is shown that clustering with hypervolume prototypes can be formulated as the minimization of an objective function second a heuristic cluster merging step is introduced where the similarity among the clusters is assessed during optimization starting with an overestimation of the number of clusters in the data similar clusters are merged in order to obtain a suitable partitioning an adaptive threshold for merging is proposed the extensions proposed are applied to gustafson kessel and fuzzy c means algorithms and the resulting extended algorithm is given the properties of the new algorithm are illustrated by various examples

on the uniqueness of lp minkowski problems the constant p curvature case in r3 abstract we study the c 4 smooth convex bodies k r n 1 satisfying k x u x 1 p where x s n k is the gauss curvature of k u is the support function of k and p is a constant in the case of n 2 either when p 1 0 or when p 0 1 in addition to a pinching condition we show that k must be the unit ball this partially answers a conjecture of lutwak yang and zhang about the uniqueness of the l p minkowski problem in r 3 moreover we give an explicit pinching constant depending only on p when p 0 1

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

brunn minkowski and prekopa leindler s inequalities under projection assumptions abstract brunn minkowski s theorem says that vol 1 k l 1 n for k l convex bodies is a concave function in and assuming a common hyperplane projection of k and l it is known that the volume itself is concave the a priori natural hypothesis of a common n k plane projection of the sets turned out in the end not to imply the 1 k th concavity of the volume function in this paper we show which is the somehow best projection type assumption that is needed in order to get concavity for vol 1 k l 1 k characterizing also the equality case in the corresponding inequality moreover we consider the same problem for its functional analogue the prekopa leindler inequality

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

comparing fuzzy partitions a generalization of the rand index and related measures in this paper we introduce a fuzzy extension of a class of measures to compare clustering structures namely measures that are based on the number of concordant and the number of discordant pairs of data points this class includes the well known rand index but also commonly used alternatives such as the jaccard measure in contrast with previous proposals our extension exhibits desirable metrical properties apart from elaborating on formal properties of this kind we present an experimental study in which we compare different fuzzy extensions of the rand index and the jaccard measure

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

valuations on polytopes containing the origin in their interiors we give a classification of non negative or borel measurable sl d invariant homogeneous valuations on the space of d dimensional convex polytopes containing the origin in their interiors the only examples are volume volume of the polar body and the euler characteristic

gaussian brunn minkowski inequalities a detailed investigation is undertaken into brunn minkowski type inequalities for gauss measure a gaussian dual brunn minkowski inequality is proved together with precise equality conditions and shown to be best possible from several points of view a possible new gaussian brunn minkowski inequality is proposed and proved to be true in some special cases throughout the study attention is paid to precise equality conditions and conditions on the coecients of dilatation interesting links are found to the s inequality and the b conjecture an example is given to show that convexity is needed in the b conjecture

affine images of isotropic measures necessary and sufficient conditions are given in order for a borel measure on the euclidean sphere to have an affine image that is isotropic a sharp reverse affine isoperimetric inequality for borel measures on the sphere is presented this leads to sharp reverse affine isoperimetric inequalities for convex bodies 2015 international press of boston inc all rights reserved

on a reverse form of the brascamp lieb inequality we prove a reverse form of the multidimensional brascamp lieb inequality our method also gives a new way to derive the brascamp lieb inequality and is rather convenient for the study of equality cases

on the number of solutions to the discrete two dimensional l0 minkowski problem abstract our main result shows the uniqueness of planar convex polygons of given outer orientations to the sides and prescribed areas of the triangles formed by the origin with any two consecutive vertices if they exist such polygons are solutions to the discrete l 0 minkowski problem in the plane proposed in a larger generality by lutwak in his extension of the brunn minkowski theory the method exploits the equivalence between the l 0 polygons and the homothetic solutions to appropriate anisotropic crystalline flows hence as a by product we obtain a positive answer to jean taylor s conjecture on the uniqueness of polygonal shapes which are deformed homothetically under the flow by crystalline curvature without any symmetry assumptions

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

comparison of empirical and computed values of fuzzy conjunction abstract the necessity for further experiments such as in 4 for the foundation of fuzzy sets theory is shown generality of the experimental results on fuzzy conjunction in the paper by thole zimmermann and zisno 1979 4 for the other objects and language is verified for other conditions usually the minimum the product and the restricted sum operators are used as an intersection conjunction of fuzzy sets in fuzzy optimization f optimization on the basis of experiments it is shown that these operators are correct for f optimization their correctness is founded on the stability of the extremal point a decision in bellman zadeh s scheme under a positive linear mapping it is shown that these mappings exist between empirical conjunction and considered operators as a statistical result of linear regression analysis

a steiner formula in the lp brunn minkowski theory abstract we prove an analogue of the classical steiner formula for the l p affine surface area of a minkowski outer parallel body for any real parameter p we show that the classical steiner formula and the steiner formula of lutwak s dual brunn minkowski theory are special cases of this new steiner formula this new steiner formula and its localized versions lead to new curvature measures that have not appeared before in the literature they have the intrinsic volumes of the classical brunn minkowski theory as well as the dual quermassintegrals of the dual brunn minkowski theory as special cases properties of these new quantities are investigated a connection to information theory among them a steiner formula for the s th mixed l p affine surface area of a minkowski outer parallel body for any real parameters p and s is also given

positions of convex bodies associated to extremal problems and isotropic measures abstract we show that there are close relations between extremal problems in dual brunn minkowski theory and isotropic type properties for some borel measures on the sphere the methods we use allow us to obtain similar results in the context of firey brunn minkowski theory we also study reverse inequalities for dual mixed volumes which are related with classical positions such as l position or isotropic position

a sensitivity analysis approach to introducing weight factors into decision functions in fuzzy multicriteria decision making in fuzzy multicriteria decision making the evaluations for criteria are aggregated using a suitable decision function many of the aggregation operators from fuzzy set theory that can be used as decision functions assume that the criteria have equal importance this paper considers the extension of these functions when the criteria are not equally weighted the criteria weights are interpreted as factors influencing the sensitivity of the decision function to the corresponding criteria a set of conditions is specified which a weighted decision function should satisfy this leads to a systematic way of introducing the weight factors into the decision functions the proposed analysis is applied to a class of operators in which the parameter may be interpreted as the decision maker s characteristic optimism index in this context it is shown that the well known method of multiplication of weight factors with membership values is suitable for optimistic decision makers while yager s power raising method is suitable for pessimistic decision makers only a geometric interpretation of the decision making process is also given

lp john ellipsoids it is shown that the classical john ellipsoid the petty ellipsoid and a recently discovered dual of the legendre ellipsoid are all special cases p infty p 1 and p 2 of a family of l p ellipsoids which can be associated with a fixed convex body this insight allows for a unified view of alternative approaches to and extensions of some basic results in convex geometry

the lp busemann petty centroid inequality abstract the ratio between the volume of the p centroid body of a convex body k in r n and the volume of k attains its minimum value if and only if k is an origin symmetric ellipsoid this result the lp busemann petty centroid inequality was recently proved by lutwak yang and zhang in this paper we show that all the intrinsic volumes of the p centroid body of k are convex functions of a time like parameter when k is moved by shifting all the chords parallel to a fixed direction the lp busemann petty centroid inequality is a consequence of this general fact

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

on the stability of brunn minkowski type inequalities we establish the stability near a euclidean ball of two conjectured inequalities the dimensional brunn minkowski inequality for radially symmetric log concave measures in rn and of the log brunn minkowski inequality

the surface measure and cone measure on the sphere of ell_p n we prove a concentration inequality for the l n p norm on the l n p sphere for p q 0 this inequality which generalizes results of schechtman and zinn 2000 is used to study the distance between the cone measure and surface measure on the sphere of l n p in particular we obtain a significant strengthening of the inequality derived by naor and romik 2003 and calculate the precise dependence of the constants that appeared there on p

product measure steady states of generalized zero range processes we establish necessary and sufficient conditions for the existence of factorizable steady states of the generalized zero range process on a periodic or infinite lattice this process allows transitions from a site i to a site i q involving a bounded number of multiple particles with rates depending on the content of the site i the direction q of movement and the number of particles moving we also show the sufficiency of a similar condition for the continuous time mass transport process where the mass at each site and the amount transferred in each transition are continuous variables we conjecture that this is also a necessary condition

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

on the lp dual minkowski problem abstract the l p dual curvature measure was introduced by lutwak yang zhang in an attempt to unify the l p brunn minkowski theory and the dual brunn minkowski theory the characterization problem for l p dual curvature measure called the l p dual minkowski problem is a fundamental problem in this unifying theory the l p dual minkowski problem contains the l p minkowski problem and the dual minkowski problem two major problems in modern convex geometry that remain open in general in this paper existence results on the l p dual minkowski problem in the weak sense will be provided moreover existence and uniqueness of the solution in the smooth category will also be demonstrated

convex bodies and norms associated to convex measures isotropy like properties are considered for finite measures with heavy tails as a basic tool we extend k ball s relationship between convex bodies and finite logarithmically concave measures to a larger class of distributions satisfying convexity conditions of the brunn minkowski type

invariances in variance estimates we provide variants and improvements of the brascamp lieb variance inequality which takes into account the invariance properties of the underlying measure this is applied to spectral gap estimates for log concave measures with many symmetries and to non interacting conservative spin systems

factorized steady states in mass transport models on an arbitrary graph we study a general mass transport model on an arbitrary graph consisting of l nodes each carrying a continuous mass the graph also has a set of directed links between pairs of nodes through which a stochastic portion of mass chosen from a site dependent distribution is transported between the nodes at each time step the dynamics conserves the total mass and the system eventually reaches a steady state this general model includes as special cases various previously studied models such as the zero range process and the asymmetric random average process we derive a general condition on the stochastic mass transport rules valid for arbitrary graph and for both parallel and random sequential dynamics that is sufficient to guarantee that the steady state is factorizable we demonstrate how this condition can be achieved in several examples we show that our generalized result contains as a special case the recent results derived by greenblatt and lebowitz for d dimensional hypercubic lattices with random sequential dynamics

geometric measures in the dual brunn minkowski theory and their associated minkowski problems a longstanding question in the dual brunn minkowski theory is what are the dual analogues of federer s curvature measures for convex bodies the answer to this is provided this leads naturally to dual versions of minkowski type problems what are necessary and sufficient conditions for a borel measure to be a dual curvature measure of a convex body sufficient conditions involving measure concentration are established for the existence of solutions to these problems

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 mixture operators using weighting functions a comparative study with wa and owa abstract in the context of multiple attribute decision making we present an aggregation scheme based on generalized mixture operators using weighting functions and we compare it with two standard aggregation methods weighted averaging and ordered weighted averaging specifically we consider linear and quadratic weight generating functions that penalize bad attribute performances and reward good attribute performances an illustrative example borrowed from the literature is used to perform the operators comparison we believe that this comparative study will highlight the potential and flexibility of generalized mixture operators using weighting functions that depend on attribute performances

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

the lp minkowski problem for polytopes for 0 p 1 abstract necessary and sufficient conditions are given for the existence of solutions to the discrete l p minkowski problem for the critical case where 0 p 1

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

cone volume measure and stability abstract we prove stability results for two central inequalities involving the cone volume measure of a centered convex body the subspace concentration conditions and the u functional volume inequality

on the planar dual minkowski problem abstract in this paper we resolve the planar dual minkowski problem proposed by huang et al 2016 31 for all positive indices without any symmetry assumption more precisely given any q 0 and function f on s 1 bounded by two positive constants we show that there exists a convex body in the plane containing the origin in its interior whose dual curvature measure c q has density f in particular if f is smooth then is also smooth

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

projection problems for symmetric polytopes abstract let k r n be a convex body a compact convex subset with non empty interior k its projection body finding the least upper bound as k ranges over the class of origin symmetric convex bodies of the affine invariant ratio v k v k n 1 being called schneider s projection problem is a well known open problem in the convex geometry to study this problem lutwak yang and zhang recently introduced a new affine invariant functional for convex polytopes in r n for origin symmetric convex polytopes they posed a conjecture for the new functional u p in this paper we give an affirmative answer to the conjecture in r n thereby obtain a modified version of schneider s projection problem

aggregation with generalized mixture operators using weighting functions this paper regards weighted aggregation operators in multiple attribute decision making and its main goal is to investigate ways in which weights can depend on the satisfaction degrees of the various attributes criteria we propose and discuss two types of weighting functions that penalize poorly satisfied attributes and reward well satisfied attributes we discuss in detail the characteristics and properties of both functions moreover we present an illustrative example to clarify the use and behaviour of such weighting functions comparing the results with those of standard weighted averaging operators

the lp minkowski problem for n p 1 abstract chou and wang s existence result for the l p minkowski problem on s n 1 for p n 1 and an absolutely continuous measure is discussed and extended to more general measures in particular we provide an almost optimal sufficient condition for the case p 0 1

an extension of minkowski s theorem and its applications to questions about projections for measures abstract minkowski s theorem asserts that every centered measure on the sphere which is not concentrated on a great subsphere is the surface area measure of some convex body and the surface area measure determines a convex body uniquely up to a shift in this manuscript we prove an extension of minkowski s theorem consider a measure on r n with positive degree of concavity and positive degree of homogeneity we show that a surface area measure of a convex set k weighted with respect to determines a convex body uniquely up to measure zero we also establish an existence result under natural conditions including symmetry we apply this result to extend the solution to classical shephard s problem to do this we introduce a new notion which relates projections of convex bodies to a given measure and is a generalization of the lebesgue volume of a projection

extremum problems for the cone volume functional of convex polytopes abstract lutwak yang and zhang defined the cone volume functional u over convex polytopes in r n containing the origin in their interiors and conjectured that the greatest lower bound on the ratio of this centro affine invariant u to volume v is attained by parallelotopes in this paper we give affirmative answers to the conjecture in r 2 and r 3 some new sharp inequalities characterizing parallelotopes in r n are established moreover a simplified proof for the conjecture restricted to the class of origin symmetric convex polytopes in r n is provided

on the brunn minkowski inequality for general measures with applications to new isoperimetric type inequalities in this paper we present new versions of the classical brunn minkowski inequality for dierent classes of measures and sets we show that the inequality a 1 b 1 n a 1 n 1 b 1 n holds true for an unconditional product measure with decreasing density and a pair of unconditional convex bodies a b r n we also show that the above inequality is true for any unconditional logconcave measure and unconditional convex bodiesa b r n finally we prove that the inequality is true for a symmetric log concave measure and a pair of symmetric convex sets a b r 2 which in particular settles two dimensional case of the conjecture for gaussian measure proposed in 13 in addition we deduce the 1 n concavity of the parallel volume t7 a tb brunn s type theorem and certain analogues of minkowski first inequality