the lp electrostatic q capacitary minkowski problem for general measures abstract existence of solutions to the l p minkowski problem of electrostatic q capacity for discrete measures was very recently given by g xiong j xiong and l xu jfa 2019 when 0 p 1 and 1 q 2 in this paper an existence result of solutions to this problem for general measures is proved when 0 p 1 and 1 q n which is a supplement to the previous result

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

general affine surface areas abstract two families of general affine surface areas are introduced basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for l affine surface areas are established

gauss curvature flow the fate of the rolling stones we prove firey s 1974 conjecture that convex surfaces moving by their gauss curvature become spherical as they contract to points

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

the minkowski problem for polytopes abstract the traditional solution to the minkowski problem for polytopes involves two steps first the existence of a polytope satisfying given boundary data is demonstrated in the second step the uniqueness of that polytope up to translation is then shown to follow from the equality conditions of minkowski s inequality a generalized isoperimetric inequality for mixed volumes that is typically proved in a separate context in this article we adapt the classical argument to prove both the existence theorem of minkowski and his mixed volume inequality simultaneously thereby providing a new proof of minkowski s inequality that demonstrates the equiprimordial relationship between these two fundamental theorems of convex geometry

the logarithmic minkowski problem in analogy with the classical minkowski problem necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone volume measure of the unit ball of a finite dimensional banach space

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

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

classification of limiting shapes for isotropic curve flows with a 5 0 and initial condition x p 0 xzo p this produces a family of curves yt x si t here n is the curvature and n is the outward pointing unit normal vector these equations are particularly natural in that they are isotropic equivariant under rotations in the plane and homogeneous equivariant under dilation of space if time is also scaled accordingly the main aim of this paper is to provide a complete description of the behaviour of embedded convex curves moving by equations of the form 1 1 in certain cases this description has already been provided if a 1 then equation 1 1 is the curve shortening flow for which the following holds

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

asymmetric affine lp sobolev inequalities abstract a new sharp affine l p sobolev inequality for functions on r n is established this inequality strengthens and implies the previously known affine l p sobolev inequality which in turn is stronger than the classical l p sobolev inequality

stationary states of relaxed open systems abstract several different first principle theories that are all potentially able to determine the long time limit of states of open quantum systems are examined and critically compared in the formal second order approximation because of a big parameter of the problem increasing time formal mathematical solution behind this second order exists just in the form of practically inapplicable formulae fortunately in the second order physical arguments could be given leading to a unique conclusion

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

the orlicz brunn minkowski theory a general framework additions and inequalities the orlicz brunn minkowski theory introduced by lutwak yang and zhang is a new extension of the classical brunn minkowski theory it represents a generalization of the lp brunn minkowski theory analogous to the way that orlicz spaces generalize lp spaces for appropriate convex functions 0 1 m 0 1 a new way of combining arbitrary sets in r n is introduced this operation called orlicz addition and denoted by has several desirable properties but is not associative unless it reduces to lp addition a general framework is introduced for the orlicz brunn minkowski theory that includes both the new addition and previously introduced concepts and makes clear for the rst time the relation to orlicz spaces and norms it is also shown that orlicz addition is intimately related to a natural and fundamental generalization of minkowski addition called m addition the results obtained show roughly speaking that the orlicz brunn minkowski theory is the most general possible based on an addition that retains all the basic geometrical properties enjoyed by the lp brunn minkowski theory inequalities of the brunn minkowski type are obtained both for m addition and orlicz addition the new orlicz brunn minkowski inequality implies the lp brunn minkowski in equality new orlicz minkowski inequalities are obtained that generalize the lp minkowski inequality one of these has connections with the conjectured log brunn minkowski inequal ity of lutwak yang and zhang and in fact these two inequalities together are shown to split the classical brunn minkowski inequality

dar s conjecture and the log brunn minkowski inequality in 1999 dar conjectured that there is a stronger version of the celebrated brunn minkowski inequality however as pointed out by campi gardner and gronchi in 2011 this problem seems to be open even for planar o symmetric convex bodies in this paper we give a positive answer to dar s conjecture for all planar convex bodies we also give the equality condition of this stronger inequality for planar o symmetric convex bodies the log brunn minkowski inequality was established by boroczky lutwak yang and zhang in 2012 it is stronger than the classical brunn minkowski inequality for planar o symmetric convex bodies gaoyong zhang asked if there is a general version of this inequality fortunately the solution of dar s conjecture especially the definition of dilation position inspires us to obtain a general version of the log brunn minkowski inequality as expected this inequality implies the classical brunn minkowski inequality for all planar convex bodies

orlicz centroid bodies the sharp affine isoperimetric inequality that bounds the volume of the centroid body of a star body from below by the volume of the star body itself is the busemann petty centroid inequality a decade ago the lp analogue of the classical busemannpetty centroid inequality was proved here the definition of the centroid body is extended to an orlicz centroid body of a star body and the corresponding analogue of the busemann petty centroid inequality is established for convex bodies the centroids of the intersections of an origin symmetric body with half spaces form the surface of a convex body this centroid body is a concept that dates back at least to dupin the classical affine isoperimetric inequality that relates the volume of a convex body with that of its centroid body was conjectured by blaschke and established in a landmark work by petty 53 petty s inequality became known as the busemann petty centroid inequality because in establishing his inequality petty not only made critical use of busemann s random simplex inequality but as petty stated he reinterpreted it see e g the books by gardner 12 leichtweiss 25 schneider 55 and thompson 58 for reference the concept of a centroid body had a natural extension in what became known as the lp brunn minkowski theory and its dual this theory had its origins in the early 1960s when firey see e g schneider 55 introduced his concept of lp compositions of convex bodies three decades later in 34 and 35 these firey minkowski lp combinations were shown to lead to an embryonic lp brunn minkowski theory this theory and its dual has witnessed a rapid growth see e g 1 9 17 23 26 32 34 44 46 48 54 56 57 59 62 the lp analogues of centroid bodies became a central focus within the lp brunn minkowski theory and its dual and establishing the lpanalogue of the busemann petty centroid inequality became a major goal this was accomplished by the authors of the present paper in 37 with an independent approach presented by campi and gronchi 3 the lp centroid bodies quickly became objects of interest in asymptotic mathematics subject classification mr 52a40 research supported in part by nsf grant dms 0706859 proof copy 1 not for distribution 2 erwin lutwak deane yang and gaoyong zhang geometric analysis see e g 10 11 24 49 50 51 52 and even the theory of stable distributions see e g 48 using concepts introduced by ludwig 29 haberl and schuster 21 were led to establish asymmetric versions of the lp busemann petty centroid inequality that for bodies that are not origin symmetric are stronger than the lp busemann petty centroid inequality presented in 37 and 3 the asymmetric inequalities obtained by haberl and schuster are ideally suited for non symmetric bodies this can be seen by looking at the haberl schuster version of the lp analogue of the classical blaschke santalo inequality that was presented in 46 while for origin symmetric bodies the lp extension of 46 does recover the original blaschke santalo inequality as p for arbitrary bodies only the haberl schuster version does so the works of haberl and schuster 21 see also 22 ludwig and reitzner 32 and ludwig 31 have demonstrated the clear need to move beyond the lp brunn minkowski theory to what we are calling an orlicz brunn minkowski theory this need is not only motivated by compelling geometric considerations such as those presented in ludwig and reitzner 32 but also by the desire to obtain sobolev bounds see 22 of a far more general nature this paper is the second in a series intended to develop a few of the elements of an orlicz brunn minkowski theory and its dual here we define the orlicz centroid body establish some of its basic properties and most importantly establish what we call the orlicz busemannpetty centroid inequality for orlicz centroid bodies we consider convex r 0 such that 0 0 this means that must be decreasing on 0 and increasing on 0 we require that one of these is happening strictly so i e is either strictly decreasing on 0 or strictly increasing on 0 the class of such will be denoted by c if k is a star body see section 1 for precise definitions with respect to the origin in rn with volume k and c then we define the orlicz centroid body k of k as the convex body whose support function at x rn is given by h k x inf 0 1 k k x y dy 1 where x y denotes the standard inner product of x and y in rn and the integration is with respect to lebesgue measure in rn when p t t p with p 1 then

the logarithmic minkowski problem for polytopes abstract the logarithmic minkowski problem asks for necessary and sufficient conditions for a finite borel measure on the unit sphere so that it is the cone volume measure of a convex body this problem was solved recently by boroczky lutwak yang and zhang for even measures boroczky et al 2013 8 this paper solves the case of discrete measures whose supports are in general position

the orlicz brunn minkowski inequality abstract the orlicz brunn minkowski theory originated with the work of lutwak yang and zhang in 2010 in this paper we first introduce the orlicz addition of convex bodies containing the origin in their interiors and then extend the l p brunn minkowski inequality to the orlicz brunn minkowski inequality furthermore we extend the l p minkowski mixed volume inequality to the orlicz mixed volume inequality by using the orlicz brunn minkowski inequality

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

on the equivalence of some projection operator techniques abstract it is shown that equations of motion for mean values of operators derived from the time convolution time convolutionless generalized master equation for the density matrix and from the equations for operators from the mori tokuyama mori theory are the same provided that projectors are suitably chosen this is valid also for systems described by time dependent hamiltonians we can emphasize that the equivalence of the correponding equations remains valid when invoking approximations

the lp minkowski problem for torsion abstract in the present paper we introduce the so called l p torsion measure l p mixed torsion a hadamard variational formula for torsion under the p sum is established as an application the existence of solutions to a l p minkowski problem for torsion is proved for any fixed p 1 and p n 2 which extends a result of colesanti and fimiani 24 a l p minkowski inequality for torsion is also obtained which implies the uniqueness of the solutions for the l p minkowski problem mentioned above

disturbance decoupling by measurement feedback with stability or pole placement in this paper we solve the disturbance decoupling problem by measurement feedback and requiring stability or pole placement on the closed loop system the problem is attacked using the geometric approach through the concepts of a bmod mathcal b invariant and controllability subspaces and their duals a mid k invariant and complementary observability subspaces the solution of this problem has an interesting structure consisting of a feedback processor which decomposes into i a disturbance decoupling loop ii a disturbance input stabilization or pole placement loop and iii a controlled output stabilization or pole placement loop

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

geometric measures in the dual brunn minkowski theory and their associated minkowski problems a longstanding question in the dual brunn minkowski theory is what are the dual analogues of federer s curvature measures for convex bodies the answer to this is provided this leads naturally to dual versions of minkowski type problems what are necessary and sufficient conditions for a borel measure to be a dual curvature measure of a convex body sufficient conditions involving measure concentration are established for the existence of solutions to these problems

almost disturbance decoupling with bounded peaking this paper is concerned with a generalization of the almost disturbance decoupling problem by state feedback apart from approximate decoupling from the external disturbances to a first to be controlled output we require a second output to be uniformly bounded with respect to the accuracy of decoupling the problem is studied using the geometric approach to linear systems we introduce some new almost controlled invariant subspaces and study their geometric structure necessary and sufficient conditions for the solvability of the above problem are formulated in terms of these controlled invariant subspaces a conceptual algorithm is introduced to calculate the feedback laws needed to achieve the design purpose

orlicz projection bodies abstract minkowski s projection bodies have evolved into l p projection bodies and their asymmetric analogs these all turn out to be part of a far larger class of orlicz projection bodies the analog of the classical petty projection inequality is established for the new orlicz projection bodies

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

disturbance decoupling and output stabilization by measurement feedback a combined approach a very general control problem is solved in the context of the geometric approach to linear multivariable control theory the problem to be considered is general in the sense that it contains as a special case not only the well known disturbance decoupling problem by measurement feedback schumacher 1981 and the problem of stabilizing the output with respect to disturbances by means of state feedback hautus 1970 but also for instance the extension of the latter problem to the situation in which only the use of measurement feedback is allowed furthermore connections with and extensions of recent work by trentelman 1986 are given and fruitful use is made of a synthesis of concepts arising from the geometric and the frequency domain approaches to control theory

the lp minkowski problem for polytopes for 0 p 1 abstract necessary and sufficient conditions are given for the existence of solutions to the discrete l p minkowski problem for the critical case where 0 p 1