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

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

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

approximate independence of distributions on spheres and their stability properties let zeta be chosen at random on the surface of the p sphere in mathbb r n 0 p n x in mathbb r n sum n i 1 x i p n if p 2 then the first k components zeta 1 ldots zeta k are for k fixed in the limit as n rightarrow infty independent standard normal considering the general case p 0 the same phenomenon appears with a distribution f p in an exponential class mathscr e f p can be characterized by the distribution of quotients of sums by conditional distributions and by a maximum entropy condition these characterizations have some interesting stability properties some discrete versions of this problem and some applications to de finetti type theorems are discussed

mixed time frequency domain based robust identification in this paper we propose a new robust identification framework that combines both frequency and time domain experimental data the main result of the paper shows that the problem of obtaining a nominal model consistent with the experimental data and bounds on the identification error can be recast as a constrained finite dimensional convex optimization problem that can be efficiently solved using linear matrix inequalities techniques this approach based upon a generalized interpolation theory contains as special cases the caratheodory fejer purely time domain and nevanlinna pick purely frequency domain problems the proposed procedure interpolates the frequency and time domain experimental data while restricting the identified system to be in an a priori given class of models resulting in a nominal model consistent with both sources of data thus it is convergent and optimal up to a factor of two with respect to central algorithms

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

representations of j central j potapov functions in both nondegenerate and degenerate cases abstract let j be an m m signature matrix i e j j and j 2 i m and let d z c z 1 denote p j d the class of all j potapov functions in d i e the set of all meromorphic m m matrix valued functions f in d with j contractive values at all points of d at which f is holomorphic further denote p j 0 d the subclass of all f p j d which are holomorphic at the origin let f p j 0 d and let f w j 0 a j w j be the taylor series representation of f in some neighborhood of 0 then it was proved in b fritzsche b kirstein u raabe on the structure of j potapov sequences linear algebra appl in press that for each n n the matrix a n can be described by its position in a matrix ball depending on the sequence a j j 0 n 1 the j potapov function f is called j central if there exists some k n such that for each integer j k the matrix a j coincides with the center of the corresponding matrix ball in this paper we derive left and right quotient representations of matrix polynomials for j central j potapov functions in d moreover we obtain recurrent formulas for the matrix polynomials involved in these quotient representations

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

a parametric extension of mixed time frequency robust identification a parametric extension to the time frequency robust identification framework is presented the results can be applied to stable linear time invariant systems on which time and or frequency experiments have been performed the parametric portion of the model should be affine in the unknown parameters which includes practical applications such as flexible structures the consistency problem is cast as a constrained finite dimensional convex optimization problem that can be formulated as a linear matrix inequality the proposed procedure provides an interpolating identification algorithm convergent and optimal up to a factor of two with respect to central algorithms

evaluation of the coulomb force via the fredholm integral equation the electrostatic force between two conducting spheres is solved exactly from coulomb s law the force is evaluated in two steps i the charge distribution on each of the spheres is calculated as a function of separation distance by way of a solution to the fredholm integral equation ii the effect of the known charge distribution is integrated to obtain the electrostatic force evaluation of the electrostatic force is fast because the series expression for the charge distribution is super converging a characteristic trait for some fredholm type equations

brief paper probabilistic bounds for l 1 uncertainty model validation this paper is concerned with a mixed deterministic probabilistic model validation problem which amounts to determining the probability for a given model to reproduce given experimental data we consider an additive uncertain model in which the modelling uncertainty is characterized in time domain by the 1 induced system norm the data available for validation are input output time series and are assumed to have been corrupted by a random noise for a given uncertainty norm bound our objective is to compute the probability for an uncertainty to exist so that it may satisfy the prescribed bound and match the input output measurements while the exact computation of this probability poses a formidable task we derive its upper and lower bounds this is carried out with respect to noise sequences with gaussian distribution and more generally when only such statistical information as the expectation and covariance of the noise are known

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

infinite dimensional convex optimization in optimal and robust control theory many engineering problems can be shown to be equivalent to solving semidefinite programs sps i e convex optimization problems involving linear matrix inequalities lmis powerful computation tools are available for such problems in the finite dimensional case however the problems arising in optimal and robust control theory are often infinite dimensional and so adequate computation tools are not available the key to tackling such problems with finite computation tools is to have a primal dual formulation of the problem without duality gap in this paper we study infinite dimensional sps and present a lifting technique to recast sps as parameterized linear programs lps this enables the wealth of theoretical tools available for infinite dimensional lps to be extended to infinite dimensional sps in particular we develop some new sufficient conditions for the lack of a duality gap for infinite dimensional sps and give an exact characterization of the primal and dual problems for these cases both primal and dual problems are formed as infinite dimensional sp problems with finite truncations to each giving upper and lower bounds respectively on the exact solution to the infinite dimensional problem thus these results can form the basis of practical computation schemes for infinite dimensional problems which require only finite dimensional computation tools to illustrate the power of these tools we apply the results to two previously unsolved optimization problems namely minimizing the l sub 1 norm of a closed loop system subject to bounds on the frequency response magnitude at a finite number of points and or bounds on the h sub 2 norm

distributions for the eigenvalues of large random matrices generated from four manifolds in this paper we will first study the spectrum of certain large euclidean random matrices the entries of these matrices are functions of n random points lp norm uniformly sampled in n dimensional lp ball or lp sphere under the setting n n 0 as n and n tend to it is shown that the empirical distributions of eigenvalues of normalized euclidean random matrices generated from the above two manifolds both converge weakly to semicircle law we also investigate certain sample covariance type matrices whose entries are determined by n random positions from an n dimensional lp ellipsoid or its surface when n and n with n n 0 we prove that their empirical spectral distribution converge to the same fixed distribution

continuous time control model validation using finite experimental data the application of robust control theory requires models containing unknown bounded perturbations and unknown bounded input signals model validation is a means of assessing the applicability of a given model with respect to experimental data this paper develops a theoretical framework and a computational solution for the model validation problem in the case where the model including unknown perturbations and signals is given in the continuous time domain yet the experimental datum is a finite sampled signal the continuous nature of the unknown components is treated directly with a sampled data lifting theory this gives results which are valid for any sample period and any datum length explicit calculation of whether sufficient data for invalidation has been obtained arises naturally in this framework a common class of robust control models is treated and leads to a convex matrix optimization problem a simulation example illustrates the approach

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

model validation a connection between robust control and identification the gap between the models used in control synthesis and those obtained from identification experiments is considered by investigating the connection between uncertain models and data the model validation problem addressed is given experimental data and a model with both additive noise and norm bounded perturbations is it possible that the model could produce the observed input output data this problem is studied for the standard h sub infinity mu framework models a necessary condition for such a model to describe an experimental datum is obtained for a large class of models in the robust control framework this condition is computable as the solution of a quadratic optimization problem

on the structure of j potapov sequences abstract let j be an m m signature matrix i e j j and j 2 i m and let d z c z 1 denote p j 0 d the set of all meromorphic m m matrix valued functions f in d which are holomorphic at 0 and take j contractive values at all points of d at which f is holomorphic then it was proved in b fritzsche b kirstein u raabe on some interrelations between j potapov functions and j potapov sequences ot series in press that the taylor coefficient sequences of the functions belonging to p j 0 d are exactly the infinite j potapov sequences of complex m m matrices the main goal of this paper is to investigate the inner structure of infinite j potapov sequences it turns out that such sequences have a clear geometric structure let a j j 0 be a j potapov sequence then we will show that for each n n the matrix a n belongs to a certain matrix ball depending on the sequence a j j 0 n 1 this observation leads us to the consideration of those j potapov sequences a j j 0 for which there exists some k n such that for j k k 1 the matrix a j coincides with the center of the corresponding matrix ball we will call these j potapov sequences j central of order k it turns out that j central j potapov sequences have a recursive structure we investigate connections between j potapov sequences and their j potapov ginzburg transforms which are m m schur sequences in particular we derive formulas containing explicit interrelations between the parameters of the corresponding matrix balls an essential consequence of these formulas is the observation that the concept of centrality is invariant with respect to j potapov ginzburg transform

a brunn minkowski theory for coconvex sets of finite volume abstract let c be a closed convex cone in r n pointed and with interior points we consider sets of the form a c k where k c is a closed convex set if a has finite volume lebesgue measure then a is called a c coconvex set and k is called c close the family of c coconvex sets is closed under the addition defined by c a 1 a 2 c a 1 c a 2 we develop first steps of a brunn minkowski theory for c coconvex sets which relates this addition to the notion of volume in particular we establish the equality condition for a brunn minkowski type inequality with reversed inequality sign and introduce mixed volumes and their integral representations for c close sets surface area measures and cone volume measures can be defined in analogy to similar notions for convex bodies they are borel measures on the intersection of the unit sphere with the interior of the polar cone of c we prove a minkowski type uniqueness theorem for c close sets with equal surface area measures concerning minkowski type existence problems we give conditions for a borel measure to be either the surface area measure or the cone volume measure of a c close set these conditions are sufficient in the first case and necessary and sufficient in the second case

on an interpolation problem for j potapov functions abstract the main goal of this paper is the study of an analogue of the matricial schur problem for the potapov class p j d of j contractive meromorphic functions in the open unit disk d using an approach which is based on an analysis of j central j potapov functions we will solve this problem in the most general case moreover in the nondegenerate case we will give a description of the corresponding weyl matrix balls furthermore we will investigate the limit behaviour of the weyl matrix balls associated with the functions belonging to some particular subclass of p j d

spectral properties of distance matrices distance matrices are matrices whose elements are the relative distances between points located on a certain manifold in all cases considered here all their eigenvalues except one are non positive when the points are uncorrelated and randomly distributed we investigate the average density of their eigenvalues and the structure of their eigenfunctions the spectrum exhibits delocalized and strongly localized states that possess different power law average behaviour the exponents depend only on the dimensionality of the manifold

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

the central limit problem for convex bodies it is shown that every symmetric convex body which satisfies a kind of weak law of large numbers has the property that almost all its marginal distributions are approximately gaussian several quite broad classes of bodies are shown to satisfy the condition

on the geometry of random convex sets between polytopes and zonotopes abstract in this work we study a class of random convex sets that interpolate between polytopes and zonotopes these sets arise from considering a q t h moment q 1 of an average of order statistics of 1 dimensional marginals of a sequence of n n independent random vectors in r n we consider the random model of isotropic log concave distributions as well as the uniform distribution on an l p n sphere 1 p with respect to the cone probability measure and study the geometry of these sets in terms of the support function and mean width we provide asymptotic formulas for the expectation of these geometric functionals which are sharp up to absolute constants our model includes and generalizes the standard one for random polytopes

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

probabilistic estimates for mixed model validation problems with cal h _ infty type uncertainties a mixed deterministic probabilistic model validation problem is investigated in this technical note which consists in an additive uncertain model with model uncertainty characterized by the h norm the data available for validation are time domain experimental data corrupted by a random noise sequence our aim is to compute the probability for such an uncertain model to be validated by the data and our main results are bounds on this probability that are computable based on the distribution of chi square random variables when the noise is a gaussian variable and solvable as an lmi problem when only statistical information such as the expectation and covariance of the noise are known

lft uncertain model validation with time and frequency domain measurements in this paper the authors study a model validation problem pertaining to linear fractional uncertainty models they extend the previous validation approaches based upon either time or frequency measurements to one using simultaneously time and frequency domain data they show that this problem can be reduced to two independent convex feasibility tests each of which corresponds to the time or frequency domain data alone and each can be verified numerically using available algorithms and software programs the merit of such a mixed time and frequency domain approach therefore lies in that not only can it accommodate the two distinctively different types of measurement data simultaneously but also from a computational standpoint it will not suffer from a complexity growth beyond that associated with the existing time domain tests

validation of linear fractional uncertain models solutions via matrix inequalities a time domain approach is provided in this paper to tackle the problem of model validation pertaining to uncertain models described by linear fractional transforms both unstructured and structured dynamic uncertainties are considered it is shown that in the first case the problem can be solved by finding a feasible solution to a convex optimization problem while in the second case it amounts to solving a biaffine matrix inequality problem to which only a convex optimization based necessary condition is given

model invalidation in spl lscr sub 1 using frequency domain data in this note we study the problem of invalidating uncertain models with an additive uncertainty the problem is to check the existence of an uncertainty and a measurement noise which fit to the given model structure and the uncertainty noise description as well as the experimental data used for invalidation we consider a mixed setting in which the uncertainty is characterized in time domain by the spl lscr sub 1 induced system norm while the available data are frequency response samples of the system we show that this problem which by formulation poses an infinite dimensional primal optimization problem can be solved in a dual finite dimensional space with finitely many constraints

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

distributions of eigenvalues of large euclidean matrices generated from lp balls and spheres abstract let x 1 x n be points randomly chosen from a set g r n and f x be a function the euclidean random matrix is given by m n f x i x j 2 n n where is the euclidean distance when n is fixed and n we prove that m n the empirical distribution of the eigenvalues of m n converges to 0 for a big class of functions of f x assuming both n and n go to infinity proportionally we obtain the explicit limit of m n when g is the l p unit ball or sphere with p 1 as corollaries we obtain the limit of a n with a n d x i x j n n and d being the geodesic distance on the ordinary unit sphere in r n we also obtain the limit of a n for the euclidean distance matrix a n x i x j n n the limits are a b v where a and b are constants and v follows the marcenko pastur law the same are also obtained for other examples appeared in physics literature including exp x i x j n n and exp d x i x j n n our results partially confirm a conjecture by do and vu 14

on computational complexity of invalidating structured uncertainty models abstract in this paper we study a class of uncertainty model validation invalidation problems for linear discrete time systems we consider models described by a linear fractional transformation lft of modeling uncertainties which are structured time invariant or time varying and are bounded in h or l 1 norms the experimental data available for use in invalidation are either input output time series observations or frequency response measurements we analyze the computational complexity associated with these problems while earlier work showed that for lft models with an unstructured uncertainty the invalidation problems can be reduced to one of solving linear matrix inequalities and hence may be tackled readily using well developed numerical methods in this paper we provide a simple proof to show that in the presence of structured uncertainties they are all np hard with respect to the number of uncertainty blocks suggesting that the problems are inherently intractable from a computational standpoint additionally we also demonstrate that the problems do become tractable when the lft model description is changed to an additive one leading to lmi based invalidation tests similar to those obtained elsewhere these results indicate that the main source of the computational difficulty lies in the lft description together with the structured nature of the uncertainties

frequency domain tests for validation of linear fractional uncertain models a frequency domain approach is adopted in this paper to tackle the problem of validating uncertainty models described by linear fractional transforms this problem amounts to verifying the consistency of certain given mathematical models to experimental information obtained from a physical plant using either input output frequency domain measurements or frequency samples of the plant linear fractional models with both unstructured and structured uncertainties are considered the problem is resolved in the former case and solved approximately in the latter both results lead to tests that are readily computable via convex optimization methods and can be implemented using standard algorithms in comparison to previously available algorithms based on time domain information the main advantage of these tests is that they have a considerably lower level of computational complexity it is shown that the validation problem reduces to one of nevanlinna pick boundary interpolation and it can be solved by computing independently a sequence of convex programs of a lower dimension each of which corresponds to only one frequency sample

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

estimation of moments of sums of independent real random variables for the sum s sum x i of a sequence x i of independent symmetric or nonnegative random variables we give lower and upper estimates of moments of s the estimates are exact up to some universal constants and extend the previous results for particular types of variables x i

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

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

controller design with multiple objectives in this paper we study multi objective control problems that give rise to equivalent convex optimization problems we develop a uniform treatment of such problems by showing their equivalence to linear programming problems with equality constraints and an appropriate positive cone we present some specialized results on duality theory and we apply them to the study of three multi objective control problems the optimal l sub 1 control with time domain constraints on the response to some fixed input the mixed h sub 2 l sub 1 control problem and the l sub 1 control with magnitude constraint on the frequency response what makes these problems complicated is that they are often equivalent to infinite dimensional optimization problems the characterization of the duality relationship between the primal and dual problem allows us to derive several results these results establish connections with special convex problems linear programming or linear matrix inequality problems uncover finite dimensional structures in the optimal solution when possible and provide finite dimensional approximations to any degree of accuracy when the problem does not appear to have a finite dimensional structure to illustrate the theory and highlight its potential several numerical examples are presented

on the weyl matrix balls associated with j potapov sequences in both nondegenerate and degenerate cases abstract in this paper we discuss the weyl matrix balls corresponding to an interpolation problem for j potapov functions in the open unit disk d which was studied in 10 in particular we derive explicit expressions for the parameters of the weyl matrix balls in terms of the given data furthermore we examine the monotonicity of the sequences of the weyl semi radii finally we discuss central j potapov functions using their associated weyl matrix balls

a time domain approach to model validation in this paper we offer a novel approach to control oriented model validation problems the problem is to decide whether a postulated nominal model with bounded uncertainty is consistent with measured input output data our approach directly uses time domain input output data to validate uncertainty models the algorithms we develop are computationally tractable and reduce to generally nondifferentiable convex feasibility problems or to linear programming problems in special cases we give analytical solutions to these problems

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

superconcentration and randomized dvoretzky s theorem for spaces with 1 unconditional bases abstract let n be a sufficiently large natural number and let b be an origin symmetric convex body in r n in the l position and such that the space r n b admits a 1 unconditional basis then for any e 0 1 2 and for random c e log n log 1 e dimensional subspace e distributed according to the rotation invariant haar measure the section b e is 1 e euclidean with probability close to one this shows that the worst case dependence on e in the randomized dvoretzky theorem in the l position is significantly better than in john s position it is a previously unexplored feature which has strong connections with the concept of superconcentration introduced by s chatterjee in fact our main result follows from the next theorem let b be as before and assume additionally that b has a smooth boundary and e n b n c e n grad b 2 for a small universal constant c 0 where grad b is the gradient of b and n is the standard gaussian measure in r n then for any p 1 c log n the p th power of the norm b p is c log n superconcentrated in the gauss space

l 1 optimal feedback controllers for mimo discrete time systems the problem considered in this paper is the design of a closed loop system consisting of a mimo discrete time plant and compensator in such a way that the system is internally stable and optimally tracks all persistent bounded inputs the solution consists of two parts first the calculation of the minimum value of the performance index a weighted transfer function which is done by solving a linear programming problem and second construction of the optimal transfer function by solving a set of linear equations it is shown that in general the discrete time problem will have a rational solution and thus appears to be of considerable practical significance

time domain validation for sample data uncertainty models we consider time domain model validation for sampled data systems with h sub spl infin compatible uncertainty models these uncertainty models consist of a nominal continuous time plant model together with given bounds for system uncertainty and signal disturbances the validation problem is to determine whether or not a given discrete sampled input output data record is consistent with a postulated uncertainty model based on continuous time interpolation theorems we provide validation algorithms for unstructured additive uncertainty models that reduce to convex programming we treat both linear time invariant and linear time varying modeling uncertainty sets

the spectrum of kernel random matrices we place ourselves in the setting of high dimensional statistical inference where the number of variables p in a dataset of interest is of the same order of magnitude as the number of observations n we consider the spectrum of certain kernel random matrices in particular n x n matrices whose i j th entry is f x i x j p or f x i x j 2 p where p is the dimension of the data and x i are independent data vectors here f is assumed to be a locally smooth function the study is motivated by questions arising in statistics and computer science where these matrices are used to perform among other things nonlinear versions of principal component analysis surprisingly we show that in high dimensions and for the models we analyze the problem becomes essentially linear which is at odds with heuristics sometimes used to justify the usage of these methods the analysis also highlights certain peculiarities of models widely studied in random matrix theory and raises some questions about their relevance as tools to model high dimensional data encountered in practice

consistency of open loop experimental frequency response data with coprime factor plant models the model data consistency problem for coprime factorization considered is given some possibly noisy frequency response data obtained by running open loop experiments on a system show that these data are consistent with a given family of perturbed coprime factor models and a time domain spl lscr sub spl infin noise model in the noise free open loop case the model data consistency problem boils down to the existence of an interpolating function in spl rscr spl hscr sub spl infin that evaluates to a finite number of complex matrices at a finite number of points on the imaginary axis a theorem on boundary interpolation in spl rscr spl hscr sub spl infin is a building block that allows one to devise computationally simple necessary and sufficient tests to check if the perturbed coprime factorization is consistent with the data for standard coprime factorizations the test involves the computation of minimum norm solutions to underdetermined complex matrix equations the schmidt mirsky theorem is used in the case of special factorizations of flexible systems for spl lscr sub spl infin noise corrupting the frequency response measurements a complete solution to the open loop noisy siso problem using the structured singular value spl mu is given

eigenvalue distributions of large euclidean random matrices for waves in random media we study probability distributions of eigenvalues of hermitian and non hermitian euclidean random matrices that are typically encountered in the problems of wave propagation in random media

field theory of euclidean matrix ensembles recently a class of hermitian matrix ensembles was proposed by mezard et al in which the matrix elements depend on the euclidean separation of coordinates drawn from a random distribution matrix ensembles of this kind appear in a variety of physical contexts in this paper we formulate and investigate the spectral properties of these euclidean random matrix ensembles within the framework of a supersymmetric statistical field theory we discuss applications of these results to various model systems

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

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

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

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