hard lefschetz theorem for valuations complex integral geometry and unitarily invariant valuations we obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets a version of the hard lefschetz theorem using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations it implies new integral geometric formulas for real submanifolds in hermitian spaces generalizing the classical kinematic formulas in euclidean

laplace transforms and valuations it is proved that the classical laplace transform is a continuous valuation which is positively gl n covariant and logarithmic translation covariant conversely these properties turn out to be sufficient to characterize this transform

on extendability by continuity of valuations on convex polytopes abstract there is a well known construction of weakly continuous valuations on convex compact polytopes in r n in this paper we investigate when a special case of this construction gives a valuation which extends by continuity in the hausdorff metric to all convex compact subsets of r n it is shown that there is a necessary condition on the initial data for such an extension in the case of r 3 more explicit results are obtained

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

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

a classification of sl n invariant valuations a classification of upper semicontinuous and sl n invariant valuations on the space of n dimensional convex bodies is established as a consequence complete characterizations of cen t ro affine and l p affine surface areas are obtained the proofs make use of a new sl n shaping process for convex bodies

a characterization of affine surface area abstract we show that every upper semicontinuous and equi affine invariant valuation on the space of d dimensional convex bodies is a linear combination of affine surface area volume and the euler characteristic

sl m c equivariant and translation covariant continuous tensor valuations abstract the space of continuous sl m c equivariant m 2 and translation covariant valuations taking values in the space of real symmetric tensors on c m r 2 m of rank r 0 is completely described the classification involves the moment tensor valuation for r 1 and is analogous to the known classification of the corresponding tensor valuations that are sl 2 m r equivariant although the method of proof cannot be adapted

convolutions transforms and convex bodies the paper studies convex bodies and star bodies in bbb r n by using radon transforms on grassmann manifolds p cosine transforms on the unit sphere and convolutions on the rotation group of bbb r n it presents dual mixed volume characterizations of i intersection bodies and l p balls which are related to certain volume inequalities for cross sections of convex bodies it considers approximations of special convex bodies by analytic bodies and various finite sums of ellipsoids which preserve special geometric properties convolution techniques are used to derive formulas for mixed volumes mixed surface measures and p cosine transforms they are also used to prove characterizations of geometric functionals such as surface area and dual quermassintegrals

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

valuations on convex sets non commutative determinants and pluripotential theory abstract a new method of constructing translation invariant continuous valuations on convex subsets of the quaternionic space h n is presented in particular new examples of sp n sp 1 invariant translation invariant continuous valuations are constructed this method is based on the theory of plurisubharmonic functions of quaternionic variables developed by the author in two previous papers bull sci math 127 1 2003 1 j geom anal 13 2 2003 183

structure of the unitary valuation algebra s alesker has shown that if g is a compact subgroup of o n acting transitively on the unit sphere sn 1 then the vector space valg of continuous translation invariant g invariant convex valuations on rn has the structure of a finite dimensional graded algebra over r satisfying poincare duality we show that the kinematic formulas for g are determined by the product pairing using this result we then show that the algebra valu n is isomorphic to r s t fn 1 fn 2 where s t have degrees 2 and 1 respectively and the polynomial fi is the degree i term of the power series log 1 s t

valuations and busemann petty type problems projection and intersection bodies define continuous and gl n contravariant valuations they played a critical role in the solution of the shephard problem for projections of convex bodies and its dual version for sections the busemann petty problem we consider the question whether fkfl implies v k v l where f is a homogeneous continuous operator on convex or star bodies which is an so n equivariant valuation important previous results for projection and intersection bodies are extended to a large class of valuations

information theoretic inequalities for contoured probability distributions we show that for a special class of probability distributions that we call contoured distributions information theoretic invariants and inequalities are equivalent to geometric invariants and inequalities of bodies in euclidean space associated with the distributions using this we obtain characterizations of contoured distributions with extremal shannon and renyi entropy we also obtain a new reverse information theoretic inequality for contoured distributions

color image segmentation using adaptive mean shift and statistical model based methods in this paper we propose an unsupervised segmentation algorithm for color images based on gaussian mixture models gmms the number of mixture components is determined automatically by adaptive mean shift in which local clusters are estimated by repeatedly searching for higher density points in feature vector space for the estimation of parameters of gmms the mean field annealing expectation maximization em is employed the mean field annealing em provides a global optimal solution to overcome the local maxima problem in a mixture model by combining the adaptive mean shift and the mean field annealing em natural color images are segmented automatically without over segmentation or isolated regions the experiments show that the proposed algorithm can produce satisfactory segmentation without any a priori information

fisher information and matrix valued valuations abstract all continuous and affinely contravariant matrix valued valuations on the sobolev space w 1 2 r n are completely classified it is shown that there is a unique such valuation this valuation turns out to be the fisher information matrix

a global optimum approach for one layer neural networks the article presents a method for learning the weights in one layer feed forward neural networks minimizing either the sum of squared errors or the maximum absolute error measured in the input scale this leads to the existence of a global optimum that can be easily obtained solving linear systems of equations or linear programming problems using much less computational power than the one associated with the standard methods another version of the method allows computing a large set of estimates for the weights providing robust mean or median estimates for them and the associated standard errors which give a good measure for the quality of the fit later the standard one layer neural network algorithms are improved by learning the neural functions instead of assuming them known a set of examples of applications is used to illustrate the methods finally a comparison with other high performance learning algorithms shows that the proposed methods are at least 10 times faster than the fastest standard algorithm used in the comparison

intersection bodies and valuations all gl n covariant star body valued valuations on convex polytopes are completely classified it is shown that there is a unique nontrivial such valuation this valuation turns out to be the so called intersection operator an operator that played a critical role in the solution of the busemann petty problem

minkowski valuations under volume constraints abstract we provide a description of the space of continuous and translation invariant minkowski valuations k n k n for which there is an upper and a lower bound for the volume of k in terms of the volume of the convex body k itself although no invariance with respect to a group acting on the space of convex bodies is imposed we prove that only two types of operators appear a family of operators having only cylinders over n 1 dimensional convex bodies as images and a second family consisting essentially of 1 homogeneous operators using this description we give improvements of some known characterization results for the difference body

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

invariant valuations on star shaped sets the brunn minkowski theory of convex bodies and mixed volumes has provided many tools for solving problems involving projections and valuations of compact convex sets in euclidean space among the most beautiful results of twentieth century convexity is hadwiger s characterization theorem for the elementary mixed volumes quermassintegrals see 3 5 9 hadwiger s characterization leads to effortless proofs of numerous results in geometric convexity including mean projection formulas for convex bodies 13 p 294 and various kinematic formulas 7 12 14 15 hadwiger s theorem also provides a connection between rigid motion invariant set functions and symmetric polynomials 1 7 recently advancements have been made in a theory introduced by lutwak 8 that is dual to the brunn minkowski theory a theory tailored for dealing with analogous questions involving star shaped sets and intersections with subspaces see also 2 4 6 in the dual theory convex bodies are replaced by star shaped sets and support functions are replaced by radial functions hadwiger s characterizaton theorem is of such fundamental importance that any candidate for a dual theory must possess a dual analogue however the dual theory in its original form was not sufficiently rich to be able to accommodate a dual of hadwiger s theorem in 6 it was shown that the natural setting for the dual theory is larger than that envisioned by previous investigators by defining the dual topology on star shaped sets in terms of the l topology on the space of n integrable functions on the unit sphere the author was able to extend the dual theory to a broad class of star shaped sets called l stars many new theorems can be proved within this larger framework including a hadwiger style classification theorem for continuous valuations on star shaped sets that are homogeneous with respect to dilation in the present paper we discard the stringent requirement of homogeneity and continue with classification theorems for continuous valuations on article no ai971601

a homogeneous decomposition theorem for valuations on convex functions abstract the existence of a homogeneous decomposition for continuous and epi translation invariant valuations on super coercive functions is established continuous and epi translation invariant valuations that are epi homogeneous of degree n are classified by duality corresponding results are obtained for valuations on finite valued convex functions

valuations with crofton formula and finsler geometry valuations admitting a smooth crofton formula are studied using geometric measure theory and rumin s cohomology of contact manifolds the main technical result is a current representation of a valuation with a smooth crofton formula a geometric interpretation of alesker s product is given for such valuations as a first application in finsler geometry a short proof of the theorem of gelfand smirnov that crofton densities are projective is derived the holmes thompson volumes in a projective finsler space are studied it is shown that they induce in a natural way valuations and that the alesker product of the k dimensional and the l dimensional holmes thompson valuation is the k l dimensional holmes thompson valuation

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

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

star valuations and dual mixed volumes since its creation by brunn and minkowski what has become known as the brunn minkowski theory has provided powerful machinery to solve a broad variety of inverse problems with stereological data the machinery of the brunn minkowski theory includes mixed volumes of minkowski symmetrization techniques such as those of steiner and blaschke isoperimetric inequalities such as the brunn minkowski minkowski and aleksandrov fenchel inequalities integral transforms such as the cosine transform and important auxiliary bodies associated with these transforms such as minkowski s projection bodies schneider s recent book 22 on the brunn minkowski theory is the best available introduction to the subject while the brunn minkowski theory has proven to be of enormous value in answering inverse questions regarding projections of convex bodies onto subspaces the theory has been of little value in answering inverse questions with data regarding intersections with subspaces however recent advancements have been made in the development of a dual brunn minkowski theory 3 4 5 6 8 9 11 13 16 23 26 27 28 29 which has been tailored specifically for dealing with such questions in contrast to the brunn minkowski theory in the dual theory convex bodies are replaced by star shaped sets and projections onto subspaces are replaced by intersections with subspaces the machinery of the dual theory includes dual mixed volumes introduced by lutwak 15 16 dual isoperimetric inequalities 16 and important auxiliary bodies known as intersection bodies first introduced by busemann for special centered convex bodies in 1 later defined for star shaped sets by lutwak 16 see also 9 a comprehensive introduction to geometric tomography including the dual brunn minkowski theory may be found in gardner s book 17 unfortunately there still remain fundamental and foundational problems with the dual brunn minkowski theory one of the most beautiful and article no 0048

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

moment entropy inequalities it is shown that the product of the renyi entropies of two independent random vectors provides a sharp lower bound for the expected value of the moments of the inner product of the random vectors this new inequality contains important geometry such as extensions of one of the fundamental affine isoperimetric inequalities the blaschke santalo inequality

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

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

fourier transform and valuations abstract it is proved that the classical fourier transform is a continuous positively gl n covariant and logarithmic translation covariant valuation moreover the fourier transform is shown to be the only such valuation on complex valued functions

valuations on manifolds and rumin cohomology smooth valuations on manifolds are studied by establishing a link with the rumin de rham complex of the co sphere bundle several operations on differential forms induce operations on smooth valuations signature operator rumin laplace operator euler verdier involution and derivation operator as an application alesker s hard lefschetz theorem for even translation invariant valuations on a finite dimensional euclidean space is generalized to all translation invariant valuations the proof uses kahler identities the rumin de rham complex and spectral geometry

a novel color detection method based on hsl color space for robotic soccer competition in a robotic soccer competition the locations of the ball and the robot are recognized through the vision system in a strong dynamic environment therefore the robot and the computer need to identify the object in real time and accurately through the vision system due to the influence of the game ball under different brightnesses the color clusters of glossy materials will produce a nonlinear behavior and will cause difficulty of recognition thus we propose a novel color detection method based on hue saturation and lightness hsl color space in this paper firstly by utilizing the hue histogram the hue and saturation hs plane and the hue and lightness hl plane can be obtained respectively in addition based on the obtained information from two planes the databases can be built for decreasing the running time and increasing the recognition success rate of identifying object moreover a novel algorithm is presented for the game ball in the robotic soccer competition finally some examples under different brightnesses are illustrated to demonstrate the preciseness and the accuracy of the proposed method are better than the traditional method

detection and classification of areca nuts with machine vision in this study we present an application of neural networks and image processing techniques for detecting and classifying the quality of areca nuts defects with diseases or insects of areca nuts were segmented by a detection line dl method six geometric features i e the principle axis length the secondary axis length axis number area perimeter and compactness of the areca nut image 3 color features i e the mean gray level of an areca nut image on the r g and b bands and defects area were used in the classification procedure a back propagation neural network classifier was employed to sort the quality of areca nuts the methodology presented herein effectively works for classifying areca nuts to an accuracy of 90 9

theory of valuations on manifolds iii multiplicative structure in the general case this is the third part of a series of articles where the theory of valuations on manifolds is constructed in the second part of this series the notion of a smooth valuation on a manifold was introduced the goal of this article is to put a canonical multiplicative structure on the space of smooth valuations on general manifolds thus extending some of the affine constructions from the first author s 2004 paper and from the first part of this series

projection bodies and valuations let be the projection operator which maps every polytope to its projection body it is well known that maps the set of polytopes pn in rn into pn that it is a valuation and that for every p pn p is affinely associated to p it is shown that these properties characterize the projection operator this proves a conjecture by lutwak

hermitian integral geometry we give in explicit form the principal kinematic formula for the action of the ane unitary group on c n together with a straightforward algebraic method for computing the full array of unitary kinematic formulas expressed in terms of certain convex valuations introduced essentially by h tasaki we introduce also several other canonical bases for the algebra of unitary invariant valuations explore their interrelations and characterize in these terms the cones of positive and monotone elements

theory of valuations on manifolds ii abstract this article is the second part in the series of articles where we are developing theory of valuations on manifolds roughly speaking valuations could be thought as finitely additive measures on a class of nice subsets of a manifold which satisfy some additional assumptions the goal of this article is to introduce a notion of a smooth valuation on an arbitrary smooth manifold and establish some of the basic properties of it

moments and valuations all measurable and rm sl n covariant vector valued valuations on convex polytopes containing the origin in their interiors are completely classified the moment vector is shown to be essentially the only such valuation

general l_p affine isoperimetric inequalities sharp l p affine isoperimetric inequalities are established for the entire class of l p projection bodies and the entire class of l p centroid bodies these new inequalities strengthen the l p petty projection and the l p busemann petty centroid inequality

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