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

on the number of homomorphisms from a finite group to a general linear group abstract we study the number of homomorphisms from a finite group to a general linear group over a finite field in particular we give a generating function of such numbers then the rogers ramanujan identities are applicable

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

global solutions of the super critical 2d quasi geostrophic equation in besov spaces abstract in this paper we study the super critical 2d dissipative quasi geostrophic equation we obtain some regularization effects allowing us to prove a global well posedness result for small initial data lying in critical besov spaces constructed over lebesgue spaces l p with p 1 local results for arbitrary initial data are also given

small ball probability estimates ψ2 behavior and the hyperplane conjecture we introduce a method which leads to upper bounds for the isotropic constant we prove that a positive answer to the hyperplane conjecture is equivalent to some very strong small probability estimates for the euclidean norm on isotropic convex bodies as a consequence of our method we obtain an alternative proof of the result of j bourgain that every 2 body has bounded isotropic constant with a slightly better estimate if k is a symmetric convex body in rn such that q 2 for every sn 1 and every q 2 then lk c log where c 0 is an absolute constant

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

saturating constructions for normed spaces ii we prove several results of the following type given finite dimensional normed space v there exists another space x with log dim x o log dim v and such that every subspace or quotient of x whose dimension is not too small contains a further subspace isometric to v this sheds new light on the structure of such large subspaces or quotients resp large sections or projections of convex bodies and allows us to solve several problems stated in the 1980s by v milman

hastings s additivity counterexample via dvoretzky s theorem the goal of this note is to show that hastings s counterexample to the additivity of minimal output von neumann entropy can be readily deduced from a sharp version of dvoretzky s theorem

application of an idea of voronoĭ to john type problems abstract using an idea of vorono many john type and minimum position problems in dimension d can be transformed into more accessible geometric problems on convex subsets of the 1 2 d d 1 dimensional cone of positive definite quadratic forms in this way we prove several new john type and minimum position results and give alternative versions and extensions of known results in particular we characterize minimum ellipsoidal shells of convex bodies and in the typical case show their uniqueness and determine the contact number these results are formulated also in terms of the circumradius of convex bodies next circumscribed ellipsoids of minimum surface area of a convex body and the corresponding minimum position problem are studied then we investigate john type characterizations of minimum positions of a convex body with respect to moments and the product of a moment and the moment of the polar body the technique used in this context finally is applied to obtain corresponding results for the mean width and the surface area

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

affine moments of a random vector an affine invariant pth moment measure is defined for a random vector and used to prove sharp moment entropy inequalities that are more general and stronger than standard moment entropy inequalities

entropy and asymptotic geometry of non symmetric convex bodies abstract we extend to the general not necessarily centrally symmetric setting a number of basic results of local theory which were known before for centrally symmetric bodies and were using very essentially the symmetry in their proofs some of these extensions look surprising the main additional tool is a study of volume behavior around the centroid of the body

gaussian processes and almost spherical sections of convex bodies on presente une preuve simple avec des estimations pointues du theoreme de dvoretzky sur l existence des sections presque spheriques ayant une grande dimension dans des corps convexes arbitraires dans r n

almost euclidean sections in symmetric spaces and concentration of order statistics abstract we prove that for any symmetric n dimensional normed space e and any e 0 e contains a 1 e euclidean subspace of dimension at least c ln n ln 1 e where c is an absolute constant the proof is based on a concentration property of order statistics of random vectors with i i d coordinates

hom a g abstract in this paper we prove that for any finite group g and any finite abelian group a the number of homomorphism from a to g is divisible by the greatest common divisor of a and g this result is a generalization of an old theorem of frobenius on the number of solutions of the equation x n 1 on a finite group

finding structure with randomness probabilistic algorithms for constructing approximate matrix decompositions low rank matrix approximations such as the truncated singular value decomposition and the rank revealing qr decomposition play a central role in data analysis and scientific computing this work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low rank matrix approximation these techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets this paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions these methods use random sampling to identify a subspace that captures most of the action of a matrix the input matrix is then compressed either explicitly or implicitly to this subspace and the reduced matrix is manipulated deterministically to obtain the desired low rank factorization in many cases this approach beats its classical competitors in terms of accuracy robustness and or speed these claims are supported by extensive numerical experiments and a detailed error analysis the specific benefits of randomized techniques depend on the computational environment consider the model problem of finding the k dominant components of the singular value decomposition of an m times n matrix i for a dense input matrix randomized algorithms require bigo mn log k floating point operations flops in contrast to bigo mnk for classical algorithms ii for a sparse input matrix the flop count matches classical krylov subspace methods but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures iii for a matrix that is too large to fit in fast memory the randomized techniques require only a constant number of passes over the data as opposed to bigo k passes for classical algorithms in fact it is sometimes possible to perform matrix approximation with a single pass over the data

isoperimetry of waists and local versus global asymptotic convex geometries existence of nicely bounded sections of two symmetric convex bodies k and l implies that the intersection of random rotations of k and l is nicely bounded for l subspace this main result immediately yields the unexpected phenomenon if k has one nicely bounded section then most sections of k are nicely bounded this existence implies randomness consequence was proved independently in giannopoulos milman and tsolomitis the main result represents a new connection between the local asymptotic convex geometry study of sections of convex bodies and the global asymptotic convex geometry study of convex bodies as a whole the method relies on the new isoperimetry of waists on the sphere due to gromov

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

on dvoretzky s theorem for subspaces of lp abstract we prove that for any 2 p and for every n dimensional subspace x of l p represented on r n whose unit ball b x is in lewis position one has the following two level gaussian concentration inequality p z e z e e z c exp c min p e 2 n e n 2 p 0 e 1 where z is the standard n dimensional gaussian vector p 0 is a constant depending only on p and c c 0 are absolute constants as a consequence we show optimal lower bound on the dimension of random almost spherical sections for these spaces in particular for any 2 p and every n dimensional subspace x of l p the euclidean space l 2 k can be 1 e embedded into x with k c p min e 2 n e n 2 p where c p 0 is a constant depending only on p this improves upon the previously known estimate due to figiel lindenstrauss and milman

averages of norms and quasi norms we compute the number of summands in q averages of norms needed to approximate an euclidean norm it turns out that these numbers depend on the norm involved essentially only through the maximal ratio of the norm and the euclidean norm particular attention is given to the case q in which the average is replaced with the maxima this is closely connected with the behavior of certain families of projective caps on the sphere

a new proof of the caffarelli kohn nirenberg theorem here we give a self contained new proof of the partial regularity theorems for solutions of incompressible navier stokes equations in three spatial dimensions these results were originally due to scheffer and caffarelli kohn and nirenberg our proof is much more direct and simpler 1998 john wiley sons inc

two observations regarding embedding subsets of euclidean spaces in normed spaces abstract this paper contains two results concerning linear embeddings of subsets of euclidean space in low dimensional normed spaces the first is an improvement of the known dependence on in dvoretzky s theorem from order of 2 to order of except for log factors the second is a joint generalization of milman s version of dvoretzky s theorem and a recent generalization by klartag and mendelson of the johnson lindenstrauss lemma

partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations in this paper we study the local behavior of the solutions to the three dimensional magnetohydrodynamic equations we are interested in both the uniform gradient estimates for smooth solutions and regularity of weak solutions it is shown that in some neighborhood of x0 t0 the gradients of the velocity field u and the magnetic field b are locally uniformly bounded in l norm as long as that either the scaled local l2 norm of the gradient or the scaled local total energy of the velocity field is small and the scaled local total energy of the magnetic field is uniformly bounded these estimates indicate that the velocity field plays a more dominant role than that of the magnetic field in the regularity theory as an immediately corollary we can derive an estimates of hausdorff dimension on the possible singular set of a suitable weak solution as in the case of pure fluid various partial regularity results are obtained as consequences of our blow up estimates

explicit euclidean embeddings in permutation invariant normed spaces abstract let x be a real normed space of dimension n n with a basis e i 1 n such that the norm is invariant under coordinate permutations assume for simplicity that the basis constant is at most 2 consider any n n and 0 e 1 4 such that n c log e 1 1 log n we provide an explicit construction of a matrix that generates a 1 e embedding of l 2 n into x

when random proportional subspaces are also random quotients we discuss when a generic subspace of some fixed proportional dimension of a finite dimensional normed space can be isomorphic to a generic quotient of some proportional dimension of another space we show in theorem 4 1 that if this happens for some natural random structures then for any proportion arbitrarily close to 1 the first space has a lot of euclidean subspaces and the second space has a lot of euclidean quotients

mean width inequalities for isotropic measures in this paper we establish barthe s mean width inequalities for continuous isotropic measures by a direct approach rather than using the brascamp lieb inequality the following results are obtained among the convex hulls of the support of isotropic mea sures on s n 1 the regular simplex inscribed in the euclidean unit ball has maximal norm in the dual situation there is a reverse result for their polar bodies moreover the case of even isotropic measures is also investigated

high dimensional random sections of isotropic convex bodies we study two properties of random high dimensional sections of convex bodies in the first part of the paper we estimate the central section function k f n k1 k for random f gn k and k rn a centrally symmetric isotropic convex body this partially answers a question raised by v d milman and a pajor see v d milman a pajor isotropic positions and inertia ellipsoids and zonoids of the unit ball of a normed n dimensional space in lecture notes in math vol 1376 springer 1989 p 88 in the second part we show that every symmetric convex body has random high dimensional sections f gn k with outer volume ratio bounded by ovr k f cnn klog 1 nn k

on partial regularity problem for 3d boussinesq equations abstract in this paper we study the partial regularity of the solutions for the three dimensional boussinesq equations we first prove a criterion of local holder continuous of the suitable weak solutions of the boussinesq equations and show that one dimensional hausdorff measure of the singular point set is zero secondly we present a local uniform gradient estimate on the suitable weak solutions and assert that the local behavior of the solution can be dominated by some scaled quantities such as the scaled local l 3 norm of the velocity besides when the initial data v 0 and 0 decay sufficiently rapidly at the distribution of the regular point set of the suitable weak solutions is also considered based on it one can find that mhd equations are more similar to navier stokes equations than boussinesq equations finally we give a local regularity criterion of the suitable weak solutions near the boundary

small ball probability estimates for log concave measures we establish a small ball probability inequality for isotropic log concave probability measures there exist absolute constants c1 c2 0 such that if x is an isotropic log concave random vector in r with 2 constant bounded by b and if a is a non zero n n matrix then for every e 0 c1 and y r p ax y 2 6 e a hs 6 e c2 b a hs a op 2 where c1 c2 0 are absolute constants

pointwise estimates for marginals of convex bodies we prove a pointwise version of the multi dimensional central limit theorem for convex bodies namely let be an isotropic log concave probability measure on rn for a typical subspace e rn of dimension nc consider the probability density of the projection of onto e we show that the ratio between this probability density and the standard gaussian density in e is very close to 1 in large parts of e here c 0 is a universal constant this complements a recent result by the second named author where the total variation metric between the densities was considered

moment entropy inequalities for a random vector the pth moment matrix is defined for a real random vector generalizing the classical covariance matrix sharp inequalities relating the pth moment and renyi entropy are established generalizing the classical inequality relating the second moment and the shannon entropy the extremal distributions for these inequalities are completely characterized

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

lowm estimates on coordinate subspaces letkbe a symmetric convex body inrn it is well known that for every 0 1 there exists a subspacefofrnwith dim f 1 n such thatpf k c mkdn f where pfdenotes the orthogonal projection ontof consider a fixed coordinate system inrn we study the question whether an analogue of can be obtained when one is restricted to choosefamong the coordinate subspacesr 1 n with 1 n we prove several coordinate versions of in terms of the cotype 2 constant of the volume ratio and other parameters ofk the basic source of our estimates is an exact coordinate analogue of in the ellipsoidal case applications to the computation of the number of lattice points inside a convex body are considered throughout the paper

extremal problems for lp surface areas and john ellipsoids abstract two closely connected extremal problems in convex geometry the minimal l p surface area and the maximal l p john ellipsoid of a convex body under translation transforms are solved it is shown that there exists a unique interior point of a convex body such that they attain their extrema

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

a comparison principle for functions of a uniformly random subspace this note demonstrates that it is possible to bound the expectation of an arbitrary norm of a random matrix drawn from the stiefel manifold in terms of the expected norm of a standard gaussian matrix with the same dimensions a related comparison holds for any convex function of a random matrix drawn from the stiefel manifold for certain norms a reversed inequality is also valid

a generalized rotationally symmetric case of the centroaffine minkowski problem abstract in this paper the centroaffine minkowski problem a critical case of the l p minkowski problem in the n 1 dimensional euclidean space is studied by its variational structure and the method of blow up analyses we obtain two sufficient conditions for the existence of solutions for a generalized rotationally symmetric case of the problem

on zeta functions of modular representations of a discrete group abstract it is proved that the generating function defined by the numbers of isomorphism classes of absolutely irreducible representations of a finitely generated group g into gl d f q n for n 1 is a rational function it is also proved that the generating function defined by weighted sums over isomorphism classes of representations of g into gl d f q n for n 1 is meromorphic over both the complex numbers and the p adic complex numbers

the plank problem for symmetric bodies given a symmetric convex bodyc andn hyperplanes in an euclidean space there is a translate of a multiple ofc at least frac 1 n 1 times as large insidec whose interior does not meet any of the hyperplanes the result generalizes bang s solution of the plank problem of tarski and has applications to diophantine approximation

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

lp moments of random vectors via majorizing measures abstract for a random vector x in r n we obtain bounds on the size of a sample for which the empirical pth moments of linear functionals are close to the exact ones uniformly on a convex body k r n we prove an estimate for a general random vector and apply it to several problems arising in geometric functional analysis in particular we find a short lewis type decomposition for any finite dimensional subspace of l p we also prove that for an isotropic log concave random vector we only need n p 2 log n sample points so that the empirical pth moments of the linear functionals are almost isometrically the same as the exact ones we obtain a concentration estimate for the empirical moments the main ingredient of the proof is the construction of an appropriate majorizing measure to bound a certain gaussian process

existence of solutions to the orlicz minkowski problem abstract in this paper the orlicz minkowski problem a generalization of the classical minkowski problem is studied using the variational method we obtain a new existence result of solutions to this problem for general measures

lp minkowski problem with not necessarily positive data abstract the l p minkowski problem in the plane has been the object of a number of recent investigations here we present various existence results for 0 p 2 in the case 0 p 2 a necessary and sufficient condition for the solvability of the problem is given

a central limit theorem for convex sets we show that there exists a sequence varepsilon n searrow0 for which the following holds let k n be a compact convex set with a non empty interior let x be a random vector that is distributed uniformly in k then there exist a unit vector in n t0 and 0 such that sup a subset mathbb r left textit prob langle x theta rangle in a frac 1 sqrt 2 pi sigma int ae frac t t 0 2 2 sigma 2 dt right leq varepsilon n qquad ast where the supremum runs over all measurable sets a and where denotes the usual scalar product in n furthermore under the additional assumptions that the expectation of x is zero and that the covariance matrix of x is the identity matrix we may assert that most unit vectors satisfy with t0 0 and 1 corresponding principles also hold for multi dimensional marginal distributions of convex sets

on partial regularity results for the navier stokes equations looking at the regularity results of scheffer respectively caffarelli kohn and nirenberg from a new point of view indicates that estimates for the pressure do not play an essential role in partial regularity results for the navier stokes equations

elliptically contoured distributions given two covariance matricesr ands for a given elliptically contoured distribution we show how simple inequalities between the matrix elements imply thate r f e s f e g whenx xi1 i2 in is a multiindex vector and f x mathop min limits i 1 mathop max limits i 2 mathop min limits i 3 max x i 1 i n orf x is the indicator function of sets such as mathop cap limits i 1 mathop cup limits i 2 mathop cap limits i 3 cup x i 1 i n mathop limits lambda i 1 i n of which the well known slepian s inequality n 1 is a special case

lp minkowski valuations on polytopes abstract for 1 p ludwig haberl and parapatits classified l p minkowski valuations intertwining the special linear group with additional conditions such as homogeneity and continuity in this paper a complete classification of l p minkowski valuations intertwining the special linear group on polytopes without any additional conditions is established for p 1 including p for n 3 and p 1 there exist valuations not mentioned before

logarithmic reduction of the level of randomness in some probabilistic geometric constructions many of the surprising phenomena occurring in high dimensions are proved by use of probabilistic arguments which show the existence of organized and regular structures but do not hint as to where exactly do these structures lie it is an intriguing question whether some of them could be realized explicitly in this paper we show that the amount of randomness used can be reduced significantly in many of these questions from asymptotic convex geometry and most of the random steps can be substituted by completely explicit algorithmic steps the main tool we use is random walks on expander graphs

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

diameters of sections and coverings of convex bodies abstract we study the diameters of sections of convex bodies in r n determined by a random n n matrix either as kernels of or as images of entries of are independent random variables satisfying some boundedness conditions and typical examples are matrices with gaussian or bernoulli random variables we show that if a symmetric convex body k in r n has one well bounded k codimensional section then for any m ck random sections of k of codimension m are also well bounded where c 1 is an absolute constant it is noteworthy that in the gaussian case when determines randomness in sense of the haar measure on the grassmann manifold we can take c 1

from john to gauss john positions via dual mixed volumes abstract we investigate the optimization of the dual mixed volumes w i s k 0 s k d n where k r n is a convex body d n the euclidean ball and sk runs over all positions of k when s is linear we give necessary and sufficient conditions for k to be in extremal position in terms of a decomposition of the identity we consider affine problems and we also present an approach involving parallel sections of k which can be understood as a dual fractional kubota formula

asymptotic formulas for the diameter of sections of symmetric convex bodies sharpening work of the first two authors for every proportion 0 1 we provide exact quantitative relations between global parameters of n dimensional symmetric convex bodies and the diameter of their random n dimensional sections using recent results of gromov and vershynin we obtain an asymptotic formula for the diameter of random proportional sections

centroid bodies and the logarithmic laplace transform a unified approach abstract we unify and slightly improve several bounds on the isotropic constant of high dimensional convex bodies in particular a linear dependence on the body s 2 constant is obtained along the way we present some new bounds on the volume of l p centroid bodies and yet another equivalent formulation of bourgain s hyperplane conjecture our method is a combination of the l p centroid body technique of paouris and the logarithmic laplace transform technique of the first named author