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

the lp electrostatic q capacitary minkowski problem for general measures abstract existence of solutions to the l p minkowski problem of electrostatic q capacity for discrete measures was very recently given by g xiong j xiong and l xu jfa 2019 when 0 p 1 and 1 q 2 in this paper an existence result of solutions to this problem for general measures is proved when 0 p 1 and 1 q n which is a supplement to the previous result

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

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

gauss curvature flow the fate of the rolling stones we prove firey s 1974 conjecture that convex surfaces moving by their gauss curvature become spherical as they contract to points

the lp minkowski problem and the minkowski problem in centroaffine geometry abstract the l p minkowski problem introduced by lutwak is solved for p n 1 in the smooth category the relevant monge ampere equation 0 1 is solved for all p 1 the same equation for p 1 is also studied and solved for p n 1 1 when p n 1 the equation is interpreted as a minkowski problem in centroaffine geometry a kazdan warner type obstruction for this problem is obtained

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

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 probabilistic approach to the geometry of the lᵨⁿ ball this article investigates by probabilistic methods various geometric questions on b n p the unit ball of ln p we propose realizations in terms of independent random variables of several distributions on b n p including the normalized volume measure these representations allow us to unify and extend the known results of the sub independence of coordinate slabs in b n p as another application we compute moments of linear functionals on b n p which gives sharp constants in khinchine s inequalities on b n p and determines the 2 constant of all directions on b n p we also study the extremal values of several gaussian averages on sections of b n p including mean width and l norm and derive several monotonicity results as p varies applications to balancing vectors in l 2 and to covering numbers of polyhedra complete the exposition

the even orlicz minkowski problem the classical minkowski problem leads to the lp minkowski problem and now to the orlicz minkowski problem existence is demonstrated for the even orlicz minkowski problem a byproduct is a new approach to the solution of the classical minkowski problem

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

classification of limiting shapes for isotropic curve flows with a 5 0 and initial condition x p 0 xzo p this produces a family of curves yt x si t here n is the curvature and n is the outward pointing unit normal vector these equations are particularly natural in that they are isotropic equivariant under rotations in the plane and homogeneous equivariant under dilation of space if time is also scaled accordingly the main aim of this paper is to provide a complete description of the behaviour of embedded convex curves moving by equations of the form 1 1 in certain cases this description has already been provided if a 1 then equation 1 1 is the curve shortening flow for which the following holds

toward a general theory of information aggregation abstract information aggregation is a pervasive issue in the field of decision making in this paper we begin an attempt to develop a comprehensive theory of this process we investigate the use of a penalty function to help guide in the aggregation process it is suggested that an aggregated value which violates a given piece of data is charged a penalty the problem of aggregation can be seen as finding the aggregated value with the least penalty we show that logical inference systems are characterized by infinite penalty functions this implies that deviation from the data are not easily accepted probability theory is characterized by equal finite penalty costs nonmonotonic logics appear to be characterized by a case in which we have at least two different levels of penalties of differing magnitudes we discuss the relationship between the relevance or bearing a piece of data has on the problem and the penalty cost associated with it we see the central role of the intersection operation

lp dual curvature measures abstract a new family of geometric borel measures on the unit sphere is introduced special cases include the l p surface area measures which extend the classical surface area measure of aleksandrov and fenchel jessen and l p integral curvature which extends alkesandrov s integral curvature in the l p brunn minkowski theory it also includes the dual curvature measures which are the duals of federer s curvature measures in the dual brunn minkowski theory this partially unifies the classical theory of mixed volumes and the newer theory of dual mixed volumes

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 busemann petty problem via spherical harmonics abstract the busemann petty problem asks whether symmetric convex bodies in r n with smaller central hyperplane sections necessarily have smaller n dimensional volume the solution has recently been completed and the answer is affirmative if n 4 and negative if n 5 in this article we present a short proof of the affirmative result and its generalization using the funk hecke formula for spherical harmonics

weighted aggregation operators based on minimization minimization based aggregation operators a g d are discussed special attention is paid to weighting function g based cases related to some fixed dissimilarity function d when d 2 x y x y 2 we recognize mixture operators and we recall some sufficient conditions for g ensuring the monotonicity of a g d for d 1 x y x y and non decreasing non increasing g a g d is shown to be the upper lower median whenever a g d is an aggregation operator

subspace concentration of dual curvature measures of symmetric convex bodies we prove a tight subspace concentration inequality for the dual curvature measures of a symmetric convex body 2018 international press of boston inc all rights reserved

interior w 2 p estimates for solutions of the monge ampere equation in this work we adapt the techniques developed in c1 to prove interior estimates for solutions of perturbations of the monge ampere equation

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

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

a stochastic ahp decision making methodology for imprecise preferences abstract existing decision making methodologies like the analytic hierarchy process ahp address imprecise pairwise comparisons by modeling crisp pairwise comparisons as fuzzy sets or a single type of probability distribution e g uniform triangular however one common issue faced by decision makers dms is bounded rationality that is dms have limited cognitive powers in specifying their preferences over multiple pairwise comparisons this result to crisp as well as imprecise pairwise comparisons furthermore given the ultimate goal of imprecise ahp is to make the decision computing weights for the criteria and the alternatives from the imprecise preferences is a must hence these various types of pairwise comparisons must be modeled using a single probability distribution for easy computation of the weights in this research a beta distribution is proposed to model the varying stochastic preferences of the dm the method of moments methodology is used to fit the varying stochastic preferences of the dm into beta stochastic pairwise comparisons since it can represent a wide variety of probability distributions additionally a non linear programming model is then developed that calculates weights which maximize the preferences of the dm while maintaining a level of consistency comparison experiments are conducted using datasets collected from literature to validate the proposed methodology

on the selection of measures of distance between probability distributions abstract a notion of relative strength and weakness is defined for the comparison of measures of distance between probability distributions and is shown to be appropriate for use in the selection of distances as quasioptimal design criteria in problems such as signal selection and detector design when the preferred criteria probability of error and asymptotic relative efficiency are intractable csiszar s work on the topological properties of divergences a class of distances also investigated by ali and silvey enables one to identify strong distances and provides a justification for the experimental findings of previous investigators

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

p q mixed geominimal surface area and p q mixed affine surface area abstract in this paper the p q mixed geominimal surface area and the p q mixed affine surface area are introduced they are extensions of the l p geominimal surface area and the l p affine surface area respectively affine invariance under unimodular affine transformations and continuity for the p q mixed geominimal surface area and the p q mixed affine surface area are discussed the existence the uniqueness and the uniform boundness of the p q mixed petty bodies are obtained affine isoperimetric inequalities for the p q mixed geominimal surface area and the p q mixed affine surface area are established

rotationally symmetric solutions to the l p minkowski problem abstract in this paper we study the l p minkowski problem for p n 1 which corresponds to the critical exponent in the blaschke santalo inequality we first obtain volume estimates for general solutions then establish a priori estimates for rotationally symmetric solutions by using a kazdan warner type obstruction finally we give sufficient conditions for the existence of rotationally symmetric solutions by a blow up analysis we also include an existence result for the l p minkowski problem which corresponds to the super critical case of the blaschke santalo inequality

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

dar s conjecture and the log brunn minkowski inequality in 1999 dar conjectured that there is a stronger version of the celebrated brunn minkowski inequality however as pointed out by campi gardner and gronchi in 2011 this problem seems to be open even for planar o symmetric convex bodies in this paper we give a positive answer to dar s conjecture for all planar convex bodies we also give the equality condition of this stronger inequality for planar o symmetric convex bodies the log brunn minkowski inequality was established by boroczky lutwak yang and zhang in 2012 it is stronger than the classical brunn minkowski inequality for planar o symmetric convex bodies gaoyong zhang asked if there is a general version of this inequality fortunately the solution of dar s conjecture especially the definition of dilation position inspires us to obtain a general version of the log brunn minkowski inequality as expected this inequality implies the classical brunn minkowski inequality for all planar convex bodies

the lp dual minkowski problem for p 1 and q 0 abstract general l p dual curvature measures have recently been introduced by lutwak yang and zhang 24 these new measures unify several other geometric measures of the brunn minkowski theory and the dual brunn minkowski theory l p dual curvature measures arise from q th dual intrinsic volumes by means of alexandrov type variational formulas lutwak yang and zhang 24 formulated the l p dual minkowski problem which concerns the characterization of l p dual curvature measures in this paper we solve the existence part of the l p dual minkowski problem for p 1 and q 0 and we also discuss the regularity of the solution

smooth solutions to the l_p l p dual minkowski problem in this paper we consider the l p dual minkowski problem by geometric variational method using anisotropic gauss kronecker curvature flows we establish the existence of smooth solutions of the l p dual minkowski problem when pq ge 0 and the given data is even if f equiv 1 we show under some restrictions on p and q that the only even smooth uniformly convex solution is the unit ball

multi step ranking of alternatives in a multi criteria and multi expert decision making environment in this work we introduce a decision model in the form of a recursive aggregation algorithm that attempts to mimic a multi step ranking process of a set of alternatives in a multi criteria and multi expert decision making environment the main idea is rather intuitive each alternative is initially assigned a list of values representing the group experts opinion about the extent to which the alternative satisfies a set of given criteria then the values for each alternative are combined with the weighted mean operator according to vectors of weights one for each decision maker in the group these weights express the personal judgement of the decision makers about the relative importance of the individual criteria consequently a new vector of values is obtained for each alternative these new values are combined again with the weighted mean operator taking into account the different degrees of influence each decision maker accepts from the rest of the group the latter aggregation step is repeated again and again for each alternative until a consensus is attained

the logarithmic minkowski problem for polytopes abstract the logarithmic minkowski problem asks for necessary and sufficient conditions for a finite borel measure on the unit sphere so that it is the cone volume measure of a convex body this problem was solved recently by boroczky lutwak yang and zhang for even measures boroczky et al 2013 8 this paper solves the case of discrete measures whose supports are in general position

variational characterization for the planar dual minkowski problem abstract in this paper we give a variational analysis to the planar dual minkowski problem in the sobolev space with the new variational characterization we can deal with existence results for prescribed not necessarily positive data meanwhile functional inequalities and multiple solutions are also obtained

stability in the busemann petty and shephard problems abstract a comparison problem for volumes of convex bodies asks whether inequalities f k f l for all s n 1 imply that vol n k vol n l where k l are convex bodies in r n and f k is a certain geometric characteristic of k by linear stability in comparison problems we mean that there exists a constant c such that for every e 0 the inequalities f k f l e for all s n 1 imply that vol n k n 1 n vol n l n 1 n c e we prove such results in the settings of the busemann petty and shephard problems and their generalizations we consider the section function f k s k vol n 1 k and the projection function f k p k vol n 1 k where is the central hyperplane perpendicular to and k is the orthogonal projection of k to in these two cases we prove linear stability under additional conditions that k is an intersection body or l is a projection body respectively then we consider other functions f k which allow to remove the additional conditions on the bodies in higher dimensions

on the uniqueness and continuity of the dual area measure abstract in this paper the uniqueness of the dual minkowski problem for the dual area measure is established via the dual minkowski inequality and the dual log minkowski inequality for real q n 1 it is proved that the weak convergence of the q th dual area measure implies the convergence of the corresponding convex bodies in the hausdorff metric and that the solution to the dual minkowski problem is continuous with respect to q

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 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

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

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

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

existence and uniqueness of cauchy problem for fuzzy differential equations under dissipative conditions the existence theorem of peano for the fuzzy differential equation x t f t x t x t 0 x 0 does not hold in general except in the special case where the fuzzy number space e n d is finite dimensional 1 or f is assumed to be continuous and bounded 2 in this paper the dissipative type conditions which guarantee the existence theorem for peano are specified based on the existence theorem of approximate solutions to above cauchy problem

orlicz john ellipsoids abstract the orlicz john ellipsoids which are in the framework of the booming orlicz brunn minkowski theory are introduced for the first time it turns out that they are generalizations of the classical john ellipsoid and the evolved l p john ellipsoids the analog of ball s volume ratio inequality is established for the new orlicz john ellipsoids the connection between the isotropy of measures and the characterization of orlicz john ellipsoids is demonstrated

a localization property of viscosity solutions to the monge ampere equation and their strict convexity the purpose of this note is to show a localization property of convex viscosity solutions to the monge ampere inequality 0 1 2 n are strictly convex

on the l p monge ampere equation abstract for the case where 0 p 1 we prove the existence of solutions of the l p monge ampere equation for arbitrary measures and show that the solution is in general not unique

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

the lp minkowski problem for torsion abstract in the present paper we introduce the so called l p torsion measure l p mixed torsion a hadamard variational formula for torsion under the p sum is established as an application the existence of solutions to a l p minkowski problem for torsion is proved for any fixed p 1 and p n 2 which extends a result of colesanti and fimiani 24 a l p minkowski inequality for torsion is also obtained which implies the uniqueness of the solutions for the l p minkowski problem mentioned above

direct approach processes in group decision making using linguistic owa operators abstract in a linguistic framework several group decision making processes by direct approach are presented these processes are designed using the linguistic ordered weighted averaging lowa operator to do so first a study is made of the properties and the axiomatic of lowa operator showing the rationality of its aggregation way and secondly we present the use of lowa operator to solve group decision making problems from individuals linguistic preference relations

continuous rotation invariant valuations on convex sets the notion of valuation on convex sets can be considered as a generalization of the notion of measure which is deflned only on the class of convex compact sets it is well known that there are important and interesting examples of valuations on convex sets which are not measures in the usual sense as for example the mixed volumes basic deflnitions and some classical examples are discussed in section 2 of this paper for more detailed information we refer to the surveys mc sch and mc3 throughout this paper all the valuations are assumed to be continuous with respect to the hausdorfi metric note that the theory of valuations which are invariant or covariant with respect to translations belongs to the classical part of convex geometry there exists an explicit description of translation invariant continuous valuations on 1 and 2 due to hadwiger h1 the case of 2 is nontrivial continuous rigid motion invariant valuations on d are completely classifled by the remarkable hadwiger theorem as linear combinations of the quermassintegrals cf h2 or for a simpler proof k

a simple simulated annealing algorithm for the maximum clique problem abstract simulated annealing technique has mostly been used to solve various optimization and learning problems and it is well known that the maximum clique problem is one of the most studied np hard optimization problems owing to its numerous applications in this note a simple simulated annealing algorithm for the maximum clique problem is proposed and tested on all 80 dimacs maximum clique instances although it is simple the proposed simulated annealing algorithm is efficient on most of the dimacs maximum clique instances the simulation results show that the proposed simulated annealing algorithm outperforms a recent efficient simulated annealing algorithm proposed by xu and ma and the solutions obtained by the proposed simulated annealing algorithm have the equal quality with those obtained by a recent trust region heuristic algorithm of stanislav busygin

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

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

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

an analytic solution to the busemann petty problem on sections of convex bodies we derive a formula connecting the derivatives of parallel section functions of an origin symmetric star body in r n with the fourier transform of powers of the radial function of the body a parallel section function or ni 1 dimensional x ray gives the ni 1 dimensional volumes of all hyperplane sections of the body orthogonal to a given direction this formula provides a new characterization of intersection bodies in r n and leads to a unifled analytic solution to the busemann petty problem suppose that k and l are two origin symmetric convex bodies inr n such that the ni 1 dimensional volume of each central hyperplane section of k is smaller than the volume of the corresponding section of l is the n dimensional volume of k smaller than the volume of l in conjunction with earlier established connections between the busemann petty problem intersection bodies and positive deflnite distributions our formula shows that the answer to the problem depends on the behavior of the ni 2 nd derivative of the parallel section functions the a rmative answer to the busemann petty problem for n 4 and the negative answer for n 5 now follow from the fact that convexity controls the second derivatives but does not control the derivatives of higher orders

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

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

aggregation pheromone density based data clustering ants bees and other social insects deposit pheromone a type of chemical in order to communicate between the members of their community pheromone that causes clumping or clustering behavior in a species and brings individuals into a closer proximity is called aggregation pheromone this article presents a new algorithm called apc for clustering data sets based on this property of aggregation pheromone found in ants an ant is placed at each location of a data point and the ants are allowed to move in the search space to find points with higher pheromone density the movement of an ant is governed by the amount of pheromone deposited at different points of the search space more the deposited pheromone more is the aggregation of ants this leads to the formation of homogenous groups of data the proposed algorithm is evaluated on a number of well known benchmark data sets using different cluster validity measures results are compared with those obtained using two popular standard clustering techniques namely average linkage agglomerative and k means clustering algorithm and with an ant based method called adaptive time dependent transporter ants for clustering atta c experimental results justify the potentiality of the proposed apc algorithm both in terms of the solution clustering quality as well as execution time compared to other algorithms for a large number of data sets

new sine ellipsoids and related volume inequalities abstract corresponding to the legendre ellipsoid and the lyz ellipsoid two new sine ellipsoids are introduced in this paper these four ellipsoids are closely related in the pythagorean relation and duality several volume inequalities and the valuation properties are obtained for two new ellipsoids

treating fuzziness in subjective evaluation data this paper proposes a technique to deal with fuzziness in subjective evaluation data and applies it to principal component analysis and correspondence analysis in the existing method or techniques developed directly from it fuzzy sets are defined from some standpoint on a data space and the fuzzy parameters of the statistical model are identified with linear programming or the method of least squares in this paper we try to map the variation in evaluation data into the parameter space while preserving information as much as possible and thereby define fuzzy sets in the parameter space clearly it is possible to use the obtained fuzzy model to derive things like the principal component scores from the extension principle however with a fuzzy model which uses the extension principle the possibility distribution spreads out as the explanatory variable values increase this does not necessarily make sense for subjective evaluations such as a 5 level evaluation for instance instead of doing so we propose a method for explicitly expressing the vagueness of evaluation using certain quantities related to the eigenvalues of a matrix which specifies the fuzzy parameter spread as a numerical example we present an analysis of subjective evaluation data on local environments

nonuniqueness of solutions to the l p minkowski problem abstract in this paper we show by example that there is no uniform estimate for the l p minkowski problem as a result we obtain the nonuniqueness of solutions to the problem

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

a proposal on improved procedures for estimating task time distributions in pert abstract the objective of using the well known pert task time estimation formulas has two components i to elicit subjective time estimates from an expert and ii to translate these estimates into something that can be used as inputs to probabilistic network analysis by incorporating the more current techniques in subjective probability elicitation and distribution theory this paper proposes and justifies a practical and more logical two step procedure for achieving both components of the objective the first step uses a seven point fractile estimation procedure the second step determines the parameters of a beta and a ramberg schmeiser distribution function that minimize the mean square deviation between the distribution function and the estimated fractiles the procedure can be easily implemented by practitioners using a canned program

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

a novel approach to probability distribution aggregation today s business world is highly competitive and unpredictable so effective decision making is of primary importance however it is difficult to make effective decisions when sufficient information is not available and decision making in such situations involves a high risk of error conventional statistics based approaches to such problems are not effective because in such situations decision making is usually in the hands of a small panel of experts however the expert opinions can be represented by probability distribution functions thus such a problem reduces to the aggregation of a set of probability distribution functions to an aggregated or consensus distribution in this paper we propose a new approach to address this problem the novelties of the proposed approach include 1 the problem is formulated as an optimization problem and 2 the overlapping area between an individual expert s distribution and an aggregated distribution is taken to measure the expertise level of that expert and subsequently to determine the weight of the expert the proposed approach in this paper is illustrated by an example reported in literature handled with the delphi method which also shows the effectiveness of our approach

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

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