Gaussian measure of sections of convex bodies

In this paper we study properties of sections of convex bodies with respect to the Gaussian measure. We develop a formula connecting the Minkowski functional of a convex symmetric body K with the Gaussian measure of its sections. Using this formula we solve an analog of the Busemann-Petty problem for Gaussian measures.     

Gaussian measure of sections of dilates and translations of convex bodies

In this paper we give a solution for the Gaussian version of the Busemann–Petty problem with additional information about dilates and translations. We also discuss the size of the Gaussian measure of the hyperplane sections of the dilates of the unit cube.     

Some properties of convex hulls generated by homogeneous Poisson point processes in an unbounded convex domain

LetC(A) be the convex hull generated by a Poisson point process in an unbounded convex setA. A representation ofAC(A) as the union of curvilinear triangles with independent areas is established. In the case whenA is a cone the properties of the representation are examined more completely. It is also indicated how to simulateC(A) directly without first simulating the process itself.     

Extreme properties of quermassintegrals of convex bodies

In this paper, we establish two theorems for the quermassintegrals of convex bodies, which are the generalizations of the well-known Aleksandrov’ s projection theorem and Loomis-Whitney’ s inequality, respectively. Applying these two theorems, we obtain a number of inequalities for the volumes of projections of convex bodies. Besides, we introduce the concept of the perturbation element of a convex body, and prove an extreme property of it.     

Approximation of convex bodies by axially symmetric bodies

Let be an arbitrary planar convex body. We prove that contains an axially symmetric convex body of area at least . Also approximation by some specific axially symmetric bodies is considered. In particular, we can inscribe a rhombus of area at least in , and we can circumscribe a homothetic rhombus of area at most about . The homothety ratio is at most . Those factors and , as well as the ratio , cannot be improved.

     

Reduced convex bodies in Euclidean space—A survey

A convex body R in Euclidean space E^d is called reduced if the minimal width Δ(K) of each convex body K⊂R different from R is smaller than Δ(R). This definition yields a class of convex bodies which contains the class of complete sets, i.e., the family of bodies of constant width. Other obvious examples in E² are regular odd-gons. We know a relatively large amount on reduced convex bodies in E². Besides theorems which permit us to understand the shape of their boundaries, we have estimates of the diameter, perimeter and area. For d≥3 we do not even have tools which permit us to recognize what the boundary of R looks like. The class of reduced convex bodies has interesting applications. We present the current state of knowledge about reduced convex bodies in E^d, recall some striking related research problems, and put a few new questions.     

Blaschke- and Minkowski-endomorphisms of convex bodies

We consider maps of the family of convex bodies in Euclidean -dimensional space into itself that are compatible with certain structures on this family: A Minkowski-endomorphism is a continuous, Minkowski-additive map that commutes with rotations. For , a representation theorem for such maps is given, showing that they are mixtures of certain prototypes. These prototypes are obtained by applying the generalized spherical Radon transform to support functions. We give a complete characterization of weakly monotonic Minkowski-endomorphisms. A corresponding theory is developed for Blaschke-endomorphisms, where additivity is now understood with respect to Blaschke-addition. Using a special mixed volume, an adjoining operator can be introduced. This operator allows one to identify the class of Blaschke-endomorphisms with the class of weakly monotonic, non-degenerate and translation-covariant Minkowski-endomorphisms.

The following application is also shown: If a (weakly monotonic and) non-trivial endomorphism maps a convex body to a homothet of itself, then this body must be a ball.

     

Reconstruction of n-dimensional convex bodies from surface tensors

In this paper, we derive uniqueness and stability results for surface tensors. Further, we develop two algorithms that reconstruct shape of n-dimensional convex bodies. One algorithm requires knowledge of a finite number of surface tensors, whereas the other algorithm is based on noisy measurements of a finite number of harmonic intrinsic volumes. The derived stability results ensure consistency of the two algorithms. Examples that illustrate the feasibility of the algorithms are presented.     

Reconstruction of convex bodies from surface tensors

We present two algorithms for reconstruction of the shape of convex bodies in the two-dimensional Euclidean space. The first reconstruction algorithm requires knowledge of the exact surface tensors of a convex body up to rank s for some natural number s. When only measurements subject to noise of surface tensors are available for reconstruction, we recommend to use certain values of the surface tensors, namely harmonic intrinsic volumes instead of the surface tensors evaluated at the standard basis. The second algorithm we present is based on harmonic intrinsic volumes and allows for noisy measurements. From a generalized version of Wirtinger's inequality, we derive stability results that are utilized to ensure consistency of both reconstruction procedures. Consistency of the reconstruction procedure based on measurements subject to noise is established under certain assumptions on the noise variables.     

Limit theorems for number of diffusion processes, which did not absorb by boundaries

We have random number of independent diffusion processes with absorption on boundaries in some region at initial time t = 0. The initial numbers and positions of processes in region is defined by the Poisson random measure. It is required to estimate the number of the unabsorbed processes for the fixed time τ > 0. The Poisson random measure depends on τ and τ → ∞. This research was partially supported by the Ministry of Education and Science of Ukraine, project 01.07/103 and University of Salerno, Italy.     

On convex decompositions of a planar point set

Let P be a planar point set in general position. Neumann-Lara et al. showed that there is a convex decomposition of P with at most elements. In this paper, we improve this upper bound to .     

Higher-order spectral analysis and weak asymptotic stability of convex processes

This paper deals with the asymptotic stability analysis of a discrete dynamical inclusion whose right-hand side is a convex process. We provide necessary and sufficient conditions for weak asymptotic stability, and obtain sharp estimates for the asymptotic null-controllability set. These estimates involve not only standard, but also higher-order spectral information on the convex process and its adjoint.     

Convolutions and multiplier transformations of convex bodies

Rotation intertwining maps from the set of convex bodies in into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on Blaschke-Minkowski homomorphisms. We show that such maps are represented by a spherical convolution operator. An application of this representation is a complete classification of all even Blaschke-Minkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator. Among further applications is the following result: If an even Blaschke-Minkowski homomorphism maps a convex body to a polytope, then it is a constant multiple of the projection body operator.

     

Limit theorems for the convex hull of random points in higher dimensions

We give a central limit theorem for the number of vertices of the convex hull of independent and identically distributed random vectors, being sampled from a certain class of spherically symmetric distributions in that includes the normal family. Furthermore, we prove that, among these distributions, the variance of exhibits the same order of magnitude as the expectation as The main tools are Poisson approximation of the point process of vertices of the convex hull and (sub/super)-martingales.

     

A note on the strong polynomiality of convex quadratic programming

We prove that a general convex quadratic program (QP) can be reduced to the problem of finding the nearest point on a simplicial cone inO(n ³ +n logL) steps, wheren andL are, respectively, the dimension and the encoding length of QP. The proof is quite simple and uses duality and repeated perturbation. The implication, however, is nontrivial since the problem of finding the nearest point on a simplicial cone has been considered a simpler problem to solve in the practical sense due to its special structure. Also we show that, theoretically, this reduction implies that (i) if an algorithm solves QP in a polynomial number of elementary arithmetic operations that is independent of the encoding length of data in the objective function then it can be used to solve QP in strongly polynomial time, and (ii) ifL is bounded by a first order exponential function ofn then (i) can be stated even in stronger terms: to solve QP in strongly polynomial time, it suffices to find an algorithm running in polynomial time that is independent of the encoding length of the quadratic term matrix or constraint matrix. Finally, based on these results, we propose a conjecture.corresponding author. The research was done when the author was at the Department of IE & OR, University of California at Berkeley, and partially supported by ONR grant N00014-91-j-1241.     

A parametric smooth variational principle and support properties of convex sets and functions

We show a modified version of Georgiev's parametric smooth variational principle, and we use it to derive new support properties of convex functions and sets. For example, our results imply that, for any proper l.s.c. convex nonaffine function h on a Banach space Y, D(∂h) is pathwise connected and R(∂h) has cardinality at least continuum. If, in addition, Y is Fréchet-smooth renormable, then R(∂h) is pathwise connected and locally pathwise connected. Analogous properties for support points and normalized support functionals of closed convex sets are proved; they extend and strengthen recent results proved by C. De Bernardi and the author for bounded closed convex sets.     

Thinning of point processes,revisited

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Approximation of convex bodies by multiple objective optimization and an application in reachable sets

In this paper, we focus on approximating convex compact bodies. For a convex body described as the feasible set in objective space of a multiple objective programme, we show that finding it is equivalent to finding the non-dominated set of a multiple objective programme. This equivalence implies that convex bodies can be approximated using multiple objective optimization algorithms. Therefore, we propose a revised outer approximation algorithm for convex multiple objective programming problems to approximate convex bodies. Finally, we apply the algorithm to solve reachable sets of control systems and use numerical examples to show the effectiveness of the algorithm.     

AN EXTENSION OF PREDICTOR-CORRECTOR ALGORITHM TO A CLASS OF CONVEX SEPARABLE PROGRAM

1.IntroductionIn[1]Mizuno,ToddandYepresentedapredictor-correctoralgorithmforlinearpramgrammingwhichpossessesaquadraticconvergencerateofthedualgaptozero.GuoandWul6]gaveamodificationofthisalgorithmforsolvingconvexquadraticprogramwithupperbounds.Itisshownthatthemodifiedmethodnotonlypreservesalltheoriginalmerits,butalsoreducesthedualgapbyaconstantfactorineachcorrectorstep,incontrasttotheMizuno,TOddandYe'soriginalpredictor--correctormethodwherethedualgapremainsunchanged.Thealgorithmdiscussedint…