inequalities between projection functions of convex bodies we obtain injectivity results for certain integral transforms on grassmann manifolds these are used to solve the intermediate cases of a problem of shephard and to give other applications to projection functions of convex bodies of revolution

nonparametric estimation of a multivariate distribution in the presence of censoring summary this paper presents examples of situations in which one wishes to estimate a multivariate distribution from data that may be right censored a distinction is made between what we term homogeneous and heterogeneous censoring it is shown how a multivariate empirical survivor function must be constructed in order to be considered a nonparametric maximum likelihood estimate of the underlying survivor function a closed form solution similar to the product limit estimate of kaplan and meier is possible with homogeneous censoring but an iterative method such as the em algorithm is required with heterogeneous censoring an example is given in which an anomaly is produced if censored multivariate data are analyzed as a series of univariate variables this anomaly is shown to disappear if

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

nonparametric estimation of successive duration times under dependent censoring summary this paper considers nonparametric inference for duration times of two successive events since the second duration process becomes observable only if the first event has occurred the length of the first duration affects the probability of the second duration being censored dependent censoring arises if the two duration times are correlated which is often the case standard approaches to this problem fail because of dependent censoring mechanism a new product limit estimator for the second duration variable and a pathdependent joint survival function estimator are proposed both modified for the dependent censoring properties of the estimators are discussed an example from lawless 1982 is studied for illustrative purposes as well as a simulation study

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

a new filled function method for global optimization a new definition of the filled function is presented in this paper based on the definition a new filled function is presented and a global optimization algorithm is developed the implementation of the algorithm on several test problems is reported with satisfactory numerical results the numerical experiments further show that the algorithm may solve higher dimensional global optimization problems

metaheuristic vs deterministic global optimization algorithms the univariate case abstract many practical problems involve the search for the global extremum in the space of the system parameters the functions to be optimized are often highly multiextremal black box with unknown analytical representations and hard to evaluate even in the case of one parameter to be adjusted in the presence of non linear constraints the interest of both the stochastic in particular metaheuristic and mathematical programming in particular deterministic communities to the comparison of metaheuristic and deterministic classes of methods is well recognized although both the communities have a huge number of journal and proceedings papers a few of them are really dedicated to a systematic comparison of the methods belonging to these two classes this paper meets the requirement of such a comparison between nature inspired metaheuristic and deterministic algorithms more than 125 000 launches of the methods have been performed and presents an attempt beneficial to practical fields including engineering design to bring together two rather disjoint communities of metaheuristic and mathematical programming researchers and applied users

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

estimation of a bivariate distribution function under random censorship summary estimators are proposed of the underlying distribution function of independent bivariate random vectors which are subject to random right censorship in each coordinate the estimators satisfy the monotonicity requirements of a distribution function are uniformly strongly consistent at a rate equal to that of the empirical distribution function and are fairly simple to compute

an information global optimization algorithm with local tuning we propose an algorithm using only the values of the objective function for solving unconstrained global optimization problems this algorithm belongs to the class of the information methods introduced by strongin numerical methods in multiextremal problems nauka moskow 1978 and differs from the other algorithms of this class by the presence of local tuning which spies on the changes of the lipschitz constant of the objective function over different sectors of the search region we describe two versions of the method for solving one dimensional problems and for solving multidimensional problems using peano type space filling curves for reduction of imensionality in both cases we establish sufficient conditions of convergence to the global minimum we also report results of some numerical experiments

on the _ minkowski problem a volume normalized formulation of the lp minkowski problem is presented this formulation has the advantage that a solution is possible for all p 1 including the degenerate case where the index p is equal to the dimension of the ambient space a new approach to the lp minkowski problem is presented which solves the volume normalized formulation for even data and all p 1

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

shadows of convex bodies it is proved that if c is a convex body in rn then c has an affine image c of nonzero volume so that if p is any 1 codimensional orthogonal projection pca 1 cl n l n it is also shown that there is a pathological body k all of whose orthogonal projections have volume about vx times as large as iki n 1 n 0 introduction the problems discussed in this paper concern the areas of shadows orthogonal projections of convex bodies and to a lesser extent the surface areas of such bodies if c is a convex body in irn and 0 a unit vector p c will denote the orthogonal projection of c onto the 1 codimensional space perpendicular to 0 volumes and areas of convex bodies and their surfaces will be denoted with i 1 the relationship between shadows and surface areas of convex bodies is expressed in cauchy s well known formula for each n e n let vn be the volume of the n dimensional euclidean unit ball and let a an be the rotationally invariant probability on the unit sphere sn cauchy s formula states that if c is a convex body in rn then its surface area is ac i s p6ci dcr o vni jn i the classical isoperimetric inequality in rn states that any body has surface area at least as large as a euclidean ball of the same volume the first section of this paper is devoted to the proof of a local isoperimetric inequality showing that all bodies have large shadows rather than merely large surface area or average shadow the principal motivation for this result is its relationship to a conjecture of vaaler and the important problems surrounding it this theorem and its connection with vaaler s conjecture are described at the beginning of 1 an important role is played in the theory of convex bodies by the so called projection body of a convex body it is easily seen by considering polytopes received by the editors september 12 1989 1980 mathematics subject classification 1985 revision primary 52a20 52a40 the author was supported in part by nsf dms 8807243 1991 american mathematical society 0002 9947 91 1 00 25 per page

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

nonparametric bivariate estimation with randomly censored data summary the estimation of a bivariate distribution function with randomly censored data is considered it is assumed that the censoring occurs independently of the lifetimes and that deaths and losses which occur simultaneously can be separated two estimators are developed a reduced sample estimator and a self consistent one it is shown that the latter estimator satisfies a nonparametric likelihood function and is unique up to the final uncensored values in any dimension it jumps at the points of double deaths in both dimensions some key word8 censored data kaplan meier estimator life table product limit estimator survival estimation

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

lipschitz gradients for global optimization in a one point based partitioning scheme a global optimization problem is studied where the objective function f x is a multidimensional black box function and its gradient f x satisfies the lipschitz condition over a hyperinterval with an unknown lipschitz constant k different methods for solving this problem by using an a priori given estimate of k its adaptive estimates and adaptive estimates of local lipschitz constants are known in the literature recently the authors have proposed a one dimensional algorithm working with multiple estimates of the lipschitz constant for f x the existence of such an algorithm was a challenge for 15 years in this paper a new multidimensional geometric method evolving the ideas of this one dimensional scheme and using an efficient one point based partitioning strategy is proposed numerical experiments executed on 800 multidimensional test functions demonstrate quite a promising performance in comparison with popular direct based methods

global search based on efficient diagonal partitions and a set of lipschitz constants in the paper the global optimization problem of a multidimensional black box function satisfying the lipschitz condition over a hyperinterval with an unknown lipschitz constant is considered a new efficient algorithm for solving this problem is presented at each iteration of the method a number of possible lipschitz constants are chosen from a set of values varying from zero to infinity this idea is unified with an efficient diagonal partition strategy a novel technique balancing usage of local and global information during partitioning is proposed a new procedure for finding lower bounds of the objective function over hyperintervals is also considered it is demonstrated by extensive numerical experiments performed on more than 1600 multidimensional test functions that the new algorithm shows a very promising performance

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

local tuning and partition strategies for diagonal go methods in this paper global optimization go lipschitz problems are considered where the multi dimensional multiextremal objective function is determined over a hyperinterval an efficient one dimensional go method using local tuning on the behavior of the objective function is generalized to the multi dimensional case by the diagonal approach using two partition strategies global convergence conditions are established for the obtained diagonal geometric methods results of a wide numerical comparison show a strong acceleration reached by the new methods working with estimates of the local lipschitz constants over different subregions of the search domain in comparison with the traditional approach

a note on nonparametric estimators of the bivariate survival function under univariate censoring summary a simple nonparametric estimator is proposed for the underlying bivariate distribution of paired failure times in the presence of univariate right censoring the estimator is shown to be a particular path dependent estimator which is a generalisation of an estimator proposed by campbell foldes 1982 under univariate censoring this path dependent estimator has the smallest variance among all path dependent estimators which includes the estimator proposed by lin ying 1993

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

the spherical convex floating body abstract for a convex body on the euclidean unit sphere the spherical convex floating body is introduced the asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated this gives rise to a new spherical area measure the floating area remarkably this floating area turns out to be a spherical analogue to the classical affine surface area from affine differential geometry several properties of the floating area are established

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

bivariate estimation with right truncated data abstract bivariate estimation with survival data has received considerable attention recently however most of the work has focused on random censoring models another common feature of survival data random truncation is considered in this study truncated data may arise if the time origin of the events under study precedes the observation period in a random right truncation model one observes the iid samples of y t only if y t where y is the variable of interest and t is an independent variable that prevents the complete observation of y suppose that y x is a bivariate vector of random variables where y is subject to right truncation in this study the bivariate reverse hazard vector is introduced and a nonparametric estimator is suggested an estimator for the bivariate survival function is also proposed weak convergence and strong consistency of this estimator are established via a representation by iid variables an expression for the limiting covariance function is provided and a

index information algorithm with local tuning for solving multidimensional global optimization problems with multiextremal constraints multidimensional optimization problems where the objective function and the constraints are multiextremal non differentiable lipschitz functions with unknown lipschitz constants and the feasible region is a finite collection of robust nonconvex subregions are considered both the objective function and the constraints may be partially defined to solve such problems an algorithm is proposed that uses peano space filling curves and the index scheme to reduce the original problem to a holder one dimensional one local tuning on the behaviour of the objective function and constraints is used during the work of the global optimization procedure in order to accelerate the search the method neither uses penalty coefficients nor additional variables convergence conditions are established numerical experiments confirm the good performance of the technique

complemented brunn minkowski inequalities and isoperimetry for homogeneous and non homogeneous measures abstract elementary proofs of sharp isoperimetric inequalities on a normed space r n equipped with a measure w x d x so that w p is homogeneous are provided along with a characterization of the corresponding equality cases when p 0 and in addition w p is assumed concave the result is an immediate corollary of the borell brascamp lieb extension of the classical brunn minkowski inequality providing a new elementary proof of a recent cabre ros oton serra result when p 1 n 0 the relevant property turns out to be a novel q complemented brunn minkowski inequality 0 1 borel sets a b r n such that r n a r n b r n a 1 b r n a q 1 r n b q 1 q which we show is always satisfied by when w p is homogeneous with 1 q 1 p n in particular this is satisfied by the lebesgue measure with q 1 n this gives rise to a new class of measures which are complemented analogues of the class of convex measures introduced by borell but which have vastly different properties the resulting isoperimetric inequality and characterization of isoperimetric minimizers extends beyond the recent results of canete rosales and howe the isoperimetric and brunn minkowski type inequalities also extend to the non homogeneous setting under a certain log convexity assumption on the density finally we obtain functional sobolev and nash type versions of the studied inequalities

a general formula for eigenvalue assignment by static output feedback with application to robust design in this note generalizing an idea of a previous paper we derive a new expression of the characteristic polynomial of a linear system subject to static output feedback from this expression an explicit form of the assignment equations in the unknown elements of the feedback matrix is obtained the equations are completely general and seem much simpler than many others proposed until now in special cases a closed or almost a closed form solution is obtained the assignment equations are then used within an algorithm for global optimization to solve the problem of exactly assigning the eigenvalues of the nominal plant while maximizing the norm of perturbation matrices for which the perturbed eigenvalues remain in specified regions of the complex plane

biometrika centenary survival analysis although papers relating to survival analysis have appeared in biometrika since the very first issue the current emphasis on the topic in this journal is fairly new especially as it relates to applications in biostatistics this survey traces the development of survival analysis throughout the twentieth century as reflected in the pages of biometrika focusing primarily on work published since 1980

two methods for solving optimization problems arising in electronic measurements and electrical engineering in this paper we introduce a common problem in electronic measurements and electrical engineering finding the first root from the left of an equation in the presence of some initial conditions we present examples of electrotechnical devices analog signal filtering where it is necessary to solve it two new methods for solving this problem based on global optimization ideas are introduced the first uses the exact a priori given global lipschitz constant for the first derivative the second method adaptively estimates local lipschitz constants during the search both algorithms either find the first root from the left or determine the global minimizers in the case when the objective function has no roots sufficient conditions for convergence of the new methods to the desired solution are established in both cases the results of numerical experiments for real problems and a set of test functions are also presented

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

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

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

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

semiparametric estimation of a panel data proportional hazards model with fixed effects this paper considers a panel duration model that has a proportional hazards specification with fixed effects the paper shows how to estimate the baseline and integrated baseline hazard functions without assuming that they belong to known finite dimensional families of functions existing estimators assume that the baseline hazard function belongs to a known parametric family therefore the estimators presented here are more general than existing ones this paper also presents a method for estimating the parametric part of the proportional hazards model with dependent right censoring under which the partial likelihood estimator is inconsistent the paper presents some monte carlo evidence on the small sample performance of the new estimators

the theil sen estimator with doubly censored data and applications to astronomy abstract the theil sen estimator of the slope parameter in simple linear regression is extended to data with both the response and the covariate subject to censoring based on inverting a suitable version of kendall s statistic this estimator requires weak assumptions and is simple to compute and a simple estimate of its asymptotic variance is obtained a second extension of the theil sen estimator based on a direct estimation of the median of pairwise slopes is given these estimators are compared numerically with versions of schmitt s estimator and applied to two data sets from the recent astronomical literature

nonparametric estimation of a bivariate survival function in the presence of censoring a new family of estimators of a bivariate survival function based on censored vectors is obtained from a decomposition of the bivariate survival function these estimators are uniformly consistent under bivariate censoring and are self consistent under univariate censoring

technical communique a global optimization technique for checking parametric robustness this note investigates the possibilities offered by a novel global optimization technique in solving some minimization problems arising in control theory the technique is described and some applications to typical control problems are reported

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

efficient estimation in the bivariate censoring model and repairing npmle the npmle in the bivariate censoring model is not consistent the problem is caused by the singly censored observations in this paper we prove that the npmle based on interval censoring the singly censored observations is ecient for this reduced data and moreover if we let the width of the interval converge to zero slowly enough then the npmle is also ecient for the original data we are able to determine a lower bound for the rate at which the bandwidth should converge to zero the eciency proof uses the general identity which holds for npmle of a linear parameter in convex models as proved in van der laan 1993a

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

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

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