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

stochastic epidemics and rumours on finite random networks abstract in this paper we investigate the stochastic spread of epidemics and rumours on networks we focus on the general stochastic sir epidemic model and a recently proposed rumour model on networks in nekovee et al 2007 3 and on networks with different random structures taking into account the structure of the underlying network at the level of the degree degree correlation function using embedded markov chain techniques and ignoring density correlations between neighbouring nodes we derive a set of equations for the final size of the epidemic rumour on a homogeneous network that can be solved numerically and compare the resulting distribution with the solution of the corresponding mean field deterministic model the final size distribution is found to switch from unimodal to bimodal form indicating the possibility of substantial spread of the epidemic rumour at a threshold value that is higher than that for the deterministic model however the difference between the two thresholds decreases with the network size n following a n 1 3 behaviour we then compare results obtained by monte carlo simulation for the full stochastic model on a homogeneous network including density correlations at neighbouring nodes with those for the approximating stochastic model and show that the latter reproduces the exact simulation results with great accuracy finally further monte carlo simulations of the full stochastic model are used to explore the effects on the final size distribution of network size and structure using homogeneous networks simple random graphs and the barabasi albert scale free networks

the branching random walk and contact process on galton watson and nonhomogeneous trees we show that the branching random walk on a galton watson tree may have one or two phase transitions depending on the relative sizes of the mean degree and the maximum degree we show that there are some galton watson trees on which the branching random walk has one phase transition while the contact process has two this contradicts a conjecture of madras and schinazi we show that the contact process has only one phase transition on some trees of uniformly exponential growth and bounded degree contradicting a conjecture of pemantle

an euler relation for valuations on polytopes abstract a locally finite point set such as the set z n of integral points gives rise to a lattice of polytopes in euclidean space taking vertices from the given point set we develop the combinatorial structure of this polytope lattice and derive euler type relations for valuations on polytopes using the language of mobius inversion in this context a new family of inversion relations is obtained thereby generalizing classical relations of euler dehn sommerville and macdonald

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

rumor processes in random environment on mathbb n and on galton watson trees the aim of this paper is to study rumor processes in random environment in a rumor process a signal starts from the stations of a fixed vertex the root and travels on a graph from vertex to vertex we consider two rumor processes in the firework process each station when reached by the signal transmits it up to a random distance in the reverse firework process on the other hand stations do not send any signal but they listen for it up to a random distance the first random environment that we consider is the deterministic 1 dimensional tree mathbb n with a random number of stations on each vertex in this case the root is the origin of mathbb n we give conditions for the survival extinction on almost every realization of the sequence of stations later on we study the processes on galton watson trees with random number of stations on each vertex we show that if the probability of survival is positive then there is survival on almost every realization of the infinite tree such that there is at least one station at the root we characterize the survival of the process in some cases and we give sufficient conditions for survival extinction

new affine measures of symmetry for convex bodies abstract grunbaum introduced measures of symmetry for convex bodies that measure how far a given convex body is from a centrally symmetric one here we introduce new measures of symmetry that measure how far a given convex body is from one with enough symmetries to define these new measures of symmetry we use affine covariant points we give examples of convex bodies whose affine covariant points are far apart in particular we give an example of a convex body whose centroid and santalo point are far apart

centro affine tensor valuations abstract all measurable sl n covariant symmetric tensor valuations on convex polytopes containing the origin in their interiors are completely classified it is shown that essentially the only examples of such valuations are the moment tensor and a tensor derived from l p surface area measures this generalizes and unifies earlier results for the scalar vector and matrix valued case

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

integral geometry of unitary area measures abstract the existence of kinematic formulas for area measures with respect to any connected closed subgroup of the orthogonal group acting transitively on the unit sphere is established in particular the kinematic operator for area measures is shown to have the structure of a co product in the case of the unitary group the algebra associated to this co product is described explicitly in terms of generators and relations as a consequence a simple algorithm that yields explicit kinematic formulas for unitary area measures is obtained

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

ellipsoids and matrix valued valuations a classification is obtained of borel measurable gl n covariant symmetric matrix valued valuations on the space of n dimensional convex polytopes the only ones turn out to be the moment matrix corresponding to the classical legendre ellipsoid and the matrix corresponding to the ellipsoid recently discovered by lutwak yang and zhang a classical concept from mechanics is the legendre ellipsoid or ellipsoid of inertia 2k associated with a convex body k r it can be defined as the unique ellipsoid centered at the center of mass of k such that the ellipsoid s moment of inertia about any axis passing through the center of mass is the same as that of k if we fix a scalar product x y for x y r 2k can be defined by the moment matrix m2 k of k this is the n n matrix with coefficients

aczel chebyshev type inequality for positive linear functions abstract an aczel chebyshev type inequality and some exact inequalities like the ones of karamata for positive linear functionals are established a counterexample on chebyshev type inequality is given

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

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

generalized translation invariant valuations and the polytope algebra abstract we study the space of generalized translation invariant valuations on a finite dimensional vector space and construct a partial convolution which extends the convolution of smooth translation invariant valuations our main theorem is that mcmullen s polytope algebra is a subalgebra of the partial convolution algebra of generalized translation invariant valuations more precisely we show that the polytope algebra embeds injectively into the space of generalized translation invariant valuations and that for polytopes in general position the convolution is defined and corresponds to the product in the polytope algebra

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

rumor processes on mathbb n and discrete renewal processes we study two rumor processes on mathbb n the dynamics of which are related to an si epidemic model with long range transmission both models start with one spreader at site 0 and ignorants at all the other sites of mathbb n but differ by the transmission mechanism in one model the spreaders transmit the information within a random distance on their right and in the other the ignorants take the information from a spreader within a random distance on their left we obtain the probability of survival information on the distribution of the range of the rumor and limit theorems for the proportion of spreaders the key step of our proofs is to show that in each model the position of the spreaders on mathbb n can be related to a suitably chosen discrete renewal process

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

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

sharp inequalities between one parameter and power means for all a b 0 with a b we prove that j p a b m 1 2p 3 a b for p 2 1 2 u 1 and j p a b m 1 2p 3 a b for p 00 2 1 2 1 and the parameter 1 2p 3 in either case is the best possible here j p a b and m p a b are the one parameter and power means of order p of two positive numbers a and b respectively

survival extinction and approximation of discrete time branching random walks we consider a general discrete time branching random walk on a countable set x we relate local strong local and global survival with suitable inequalities involving the first moment matrix m of the process in particular we prove that while the local behavior is characterized by m the global behavior cannot be completely described in terms of properties involving m alone moreover we show that locally surviving branching random walks can be approximated by sequences of spatially confined and stochastically dominated branching random walks which eventually survive locally if the possibly finite state space is large enough an analogous result can be achieved by approximating a branching random walk by a sequence of multitype contact processes and allowing a sufficiently large number of particles per site we compare these results with the ones obtained in the continuous time case and we give some examples and counterexamples

generalization and sharpness of the power means inequality and their applications abstract in this paper we generalize and sharpen the power means inequality by using the theory of majorization and the analytic techniques our results unify some optimal versions of the power means inequality as application a well known conjectured inequality proposed by janous et al is proven furthermore these results are used for studying a class of geometric inequalities for simplex from which some interesting inequalities including the refined euler inequality and the reversed finsler hadwiger type inequality are obtained

subword complexity and projection bodies abstract a polytope p 0 1 d and an 0 1 d induce a so called hartman sequence h p 0 1 z which is by definition 1 at the kth position if k mod 1 p and 0 otherwise k z we prove an asymptotic formula for the subword complexity of such a hartman sequence this result establishes a connection between symbolic dynamics and convex geometry if the polytope p is convex then the subword complexity of h p asymptotically equals the volume of the projection body p of p for almost all 0 1 d

valuations on polytopes containing the origin in their interiors we give a classification of non negative or borel measurable sl d invariant homogeneous valuations on the space of d dimensional convex polytopes containing the origin in their interiors the only examples are volume volume of the polar body and the euler characteristic

the spread of a rumor or infection in a moving population we consider the following interacting particle system there is a gas of particles each of which performs a continuous time simple random walk on z d with jump rate d a these particles are called a particles and move independently of each other they are regarded as individuals who are ignorant of a rumor or are healthy we assume that we start the system with n a x 0 a particles at x and that the n a x 0 x z d are i i d mean a poisson random variables in addition there are b particles which perform continuous time simple random walks with jump rate d b we start with a finite number of b particles in the system at time 0 b particles are interpreted as individuals who have heard a certain rumor or who are infected the b particles move independently of each other the only interaction is that when a b particle and an a particle coincide the latter instantaneously turns into a b particle we investigate how fast the rumor or infection spreads specifically if b t x z d a b particle visits x during 0 t and b t b t 1 2 1 2 d then we investigate the asymptotic behavior of b t our principal result states that if d a d b so that the a and b particles perform the same random walk then there exist constants 0 c i such that almost surely c c 2 t c b t c c c 1 t for all large t where c r r r d in a further paper we shall use the results presented here to prove a full shape theorem saying that t 1 b t converges almost surely to a nonrandom set b 0 with the origin as an interior point so that the true growth rate for b t is linear in t if d a d b then we can only prove the upper bound b t c c c 1 t eventually

limit theorems for a general stochastic rumour model we study a general stochastic rumour model in which an ignorant individual has a certain probability of becoming a stifler immediately upon hearing the rumour we refer to this special kind of stifler as an uninterested individual our model also includes distinct rates for meetings between two spreaders in which both become stiflers or only one does so that particular cases are the classical daley kendall and maki thompson models we prove a law of large numbers and a central limit theorem for the proportions of those who ultimately remain ignorant and those who have heard the rumour but become uninterested in it

generalization of the power means and their inequalities abstract in this paper we gave a generalization of power means which include positive nonlinear functionals for these means we obtained generalizations of fundamental mean inequality holder s and minkowski s and their converse inequalities

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 spreading a rumor suppose that one of n people knows a rumor at the first stage he passes the rumor to someone chosen at random at each stage each person already informed knower communicates the rumor to a person chosen at random and independently of all other past and present choices denote by s n the random number of stages before everybody is informed how large is s n typically frieze and grimmet who introduced this problem proved that in probability s n log 2 n log n to 1 in this paper we show that in fact s n log 2 n log n o 1 in probability our proof demonstrates that the number i t of persons informed after t stages obeys very closely with high probability a deterministic equation i t 1 n n i t exp i t n t geqq 0 a case when each knower passes the rumor to several members at every stage is also discussed

a spatial stochastic model for rumor transmission we consider an interacting particle system representing the spread of a rumor by agents on the d dimensional integer lattice each agent may be in any of the three states belonging to the set 0 1 2 here 0 stands for ignorants 1 for spreaders and 2 for stiflers a spreader tells the rumor to any of its nearest ignorant neighbors at rate at rate a spreader becomes a stifler due to the action of other nearest neighbor spreaders finally spreaders and stiflers forget the rumor at rate one we study sufficient conditions under which the rumor either becomes extinct or survives with positive probability

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

discrete one dimensional coverage process on a renewal process consider the following coverage model on mathbb n for each site i in mathbb n associate a pair xi i r i where xi i i ge 0 is a 1 dimensional undelayed discrete renewal point process and r i i ge 0 is an i i d sequence of mathbb n valued random variables at each site where xi i 1 start an interval of length r i coverage occurs if every site of mathbb n is covered by some interval we obtain sharp conditions for both positive and null probability of coverage as corollaries we extend results of the literature of rumor processes and discrete one dimensional boolean percolation

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

characterization of critical values of branching random walks on weighted graphs through infinite type branching processes we study the branching random walk on weighted graphs site breeding and edge breeding branching random walks on graphs are seen as particular cases two kinds of survival can be identified a weak survival with positive probability there is at least one particle alive somewhere at any time and a strong survival with positive probability the colony survives by returning infinitely often to a fixed site the behavior of the process depends on the value of a certain parameter which controls the birth rates the threshold between survival and almost sure extinction is called critical value we describe the strong critical value in terms of a geometrical parameter of the graph we characterize the weak critical value and relate it to another geometrical parameter we prove that at the strong critical value the process dies out locally almost surely while at the weak critical value global survival and global extinction are both possible

convexity of lp intersection bodies we extend the classical brunn theorem to symmetric moments of convex bodies and use it to prove the convexity of the lp intersection bodies of centered convex bodies

renyi divergence and lp affine surface area for convex bodies abstract we show that the fundamental objects of the l p brunn minkowski theory namely the l p affine surface areas for a convex body are closely related to information theory they are exponentials of renyi divergences of the cone measures of a convex body and its polar we give geometric interpretations for all renyi divergences d not just for the previously treated special case of relative entropy which is the case 1 now no symmetry assumptions are needed and if at all only very weak regularity assumptions are required previously the relative entropies appeared only after performing second order expansions of certain expressions now already first order expansions make them appear thus in the new approach we detect faster details about the boundary of a convex body

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

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

on rumour propagation among sceptics junior et al j appl probab 48 624 636 2011 studied a model to understand the spread of a rumour their model consists of individuals situated at the integer points of the line mathbb n an individual at the origin 0 starts a rumour and passes it to all individuals in the interval 0 r 0 where r 0 is a non negative random variable an individual located at i in this interval receives the rumour and transmits it further among individuals in i i r i where r 0 and r i are i i d random variables the rumour spreads in this manner an alternate model considers individuals seeking to find the rumour from individuals who have already heard it for this s he asks individuals to the left of her him and lying in an interval of a random size we study these two models when the individuals are more sceptical and they transmit or accept the rumour only if they receive it from at least two different sources in stochastic geometry the equivalent of this rumour process is the study of coverage of the space mathbb n d by random sets our study here extends the study of coverage of space and considers the case when each vertex of mathbb n d is covered by at least two distinct random sets

on jessen s inequality for convex functions ii on etablit quelques resultats lies a l integralite de b jessen pour les fonctions convexes

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

some results on extending and sharpening the weierstrass product inequalities abstract in this paper we establish two extensions of weierstrass s inequality involving symmetric functions by means of the theory of majorization and give an interesting sharpness of weierstrass s inequality by using the arithmetic geometric mean inequality furthermore we apply these results to improve a well known inequality and deduce some new inequalities

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

affine vs euclidean isoperimetric inequalities abstract it is shown that every even zonal measure on the euclidean unit sphere gives rise to an isoperimetric inequality for sets of finite perimeter which directly implies the classical euclidean isoperimetric inequality the strongest member of this large family of inequalities is shown to be the only affine invariant one among them the petty projection inequality as an application a family of sharp sobolev inequalities for functions of bounded variation is obtained each of which is stronger than the classical sobolev inequality moreover corresponding families of l p isoperimetric and sobolev type inequalities are also established

two optimal double inequalities between power mean and logarithmic mean for email protected r the power mean m p a b of order p the logarithmic mean l a b and the arithmetic mean a a b of two positive real values a and b are defined by m p a b a p b p2 1 p p 0 ab p 0 l a b b alogb loga a b a a b and a a b a b2 respectively in this article we answer the questions what are the greatest values p and r and the least values q and s such that the inequalities m p a b 13a a b 23l a b m q a b and m r a b 23a a b 13l a b m s a b hold for all a b 0

majorization and relative concavity abstract in this paper some results established in h n shi refinement and generalization of a class of inequalities for symmetric functions math practice theory 29 4 1999 pp 81 84 are extended from the classical majorization preordering to group induced cone orderings to this end the notion of relative concavity introduced in c p niculescu f popovici the extension of majorization inequalities within the framework of relative convexity j inequal pure appl math 7 1 2005 article 27 is used in addition some ky fan s inequalities are discussed

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

contact measures in isotropic spaces abstract we revisit the contact measures introduced by firey and further developed by schneider and teufel from the perspective of the theory of valuations on manifolds this reveals a link between the kinematic formulas for area measures studied by wannerer and the integral geometry of curved isotropic spaces as an application we find explicitly the kinematic formula for the surface area measure in hermitian space

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

integral geometry in minkowski spaces abstract we investigate the possibility of extending classical integral geometric results involving lower dimensional areas from euclidean space to minkowski spaces finite dimensional banach spaces of the two natural notions of area in a minkowski space due respectively to busemann and to holmes and thompson the latter turns out to be the more tractable one for the holmes thompson area we derive a translative intersection formula and in the class of hypermetric minkowski spaces full analogues of the crofton formulae for rectifiable sets and for convex bodies for the busemann k area we give a short proof of the fact that it coincides for k rectifiable sets with the k dimensional hausdorff measure induced by the minkowski metric

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

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

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

classification of invariant valuations on the quaternionic plane abstract we describe the orbit space of the action of the group sp 2 sp 1 on the real grassmann manifolds gr k h 2 in terms of certain quaternionic matrices of moore rank not larger than 2 we then give a complete classification of valuations on the quaternionic plane h 2 which are invariant under the action of the group sp 2 sp 1

inequalities for convex sequences and their applications using the theory of majorization new inequalities for convex sequences are proved a necessary and sufficient condition for a convex sequence is established this reveals an important relationship between the convex sequence and the majorized inequality finally some applications of our new inequalities and some improvements of certain known results are presented

valuation theory of indefinite orthogonal groups abstract let so p q denote the identity connected component of the real orthogonal group with signature p q we give a complete description of the spaces of continuous and generalized translation and so p q invariant valuations generalizing hadwiger s classification of euclidean isometry invariant valuations as a result of independent interest we identify within the space of translation invariant valuations the class of klain schneider continuous valuations which strictly contains all continuous translation invariant valuations the operations of pull back and push forward by a linear map extend naturally to this class

relative entropies for convex bodies we introduce a new class of not necessarily convex bodies and show among other things that these bodies provide yet another link between convex geometric analysis and information theory namely they give geometric interpretations of the relative entropy of the cone measures of a convex body and its polar and related quantities such interpretations were first given by paouris and werner for symmetric convex bodies in the context of the lp centroid bodies there the relative entropies appear after performing second order expansions of certain expressions now no symmetry assumptions are needed moreover using the new bodies already first order expansions make the relative entropies appear thus these bodies detect faster details of the boundary of a convex body than the lp centroid bodies