Star-shaped Sets Research Articles

Weighted estimates for rough oscillatory singular integrals

Weighted Lp estimates (1<p<∞) are shown for oscillatory singular integral operators with polynomial phase and a rough kernel of the form eiP(x,y)Ω(x−y)h(|x−y|)|x−y|−n. We assume that Ω∈L logL(Sn−1) is homogeneous of degree zero and ∫Sn-1Ω=0. The radial factor h has bounded variation. The necessary condition on the weight is similar to the Ap condition but involves rectangles (instead of cubes) arising from a covering of a star-shaped set related to Ω.

Sets of Locally Maximal Visibility and Finite Unions of Starshaped Sets

Let \(\), where \(\) is an open connected subset of some linear topological space, such that S contains all triangular regions whose (relative) boundaries lie in S. If some finite subset T of S has locally maximal visibility in S, then \(\). Hence S is a finite union of starshaped sets whose kernels are determined by T. An analogous result holds for S open. Moreover, counterexamples show that neither the requirement on triangular regions nor the restriction to a finite set T can be deleted.

Convex intersection bodies in three and four dimensions

The paper shows that no origin-symmetric convex polyhedron in R3 is the intersection body of a star body. It is shown also that every originsymmetric convex body in Rd, for d = 3 and 4, can be seen as the intersection body of a star-shaped set whose radial function satisfies conditions related to suitable non-integer Sobolev classes.

Wedgelets: nearly minimax estimation of edges

We study a simple \Horizon Model for the problem of recovering an image from noisy data; in this model the image has an edge with fi-Holder regularity. Adopting the viewpoint of computational harmonic analysis, we develop an overcomplete collection of atoms called wedgelets, dyadically organized indicator functions with a variety of locations, scales, and orientations. The wedgelet representation provides nearly-optimal representations of objects in the Horizon model, as measured by minimax description length. We show how to rapidly compute a wedgelet approximation to noisy data by flnding a special edgelet-decorated recursive partition which minimizes a complexity-penalized sum of squares. This estimate, using su-cient sub-pixel resolution, achieves nearly the minimax mean-squared error in the Horizon Model. In fact, the method is adaptive in the sense that it achieves nearly the minimax risk for any value of the unknown degree of regularity of the Horizon, 1• fi• 2. Wedgelet analysis and de-noising may be used successfully outside the Horizon model. We study images modelled as indicators of star-shaped sets with smooth bound- aries and show that complexity-penalized wedgelet partitioning achieves nearly the minimax risk in that setting also.

Several remarks on star-shaped sets

Questions related to the properties of unions and intersections of star-shaped compact subsets of finite-dimensional spaces are considered.

Characterizing Starshaped Sets by Maximal Visibility

Let S be a nonempty set in a real topological linear space L. p ∈ S is a point of maximal visibility of S if and only if it admits a neighbourhood N in L such that Sq \(\subseteq\) Sp for every point q ∈ S ∩ N, where Sx = { s ∈ S : x is visible from s via S }. For S being either open and connected or the closure of its connected interior, it is shown that the kernel of S is the set of all maximal visibility points of S. Planar examples reveal that the topological assumptions on S are necessary. This substantially strengthens a recent result of Toranzos and Forte Cunto.

Robot motion planning on N-dimensional star worlds among moving obstacles

Inspired by an idea of Rimon and Koditschek (1992), we develop a motion planning algorithm for a point robot traveling among moving obstacles in an N-dimensional space. The navigating point must meet a goal point at a fixed time T, while avoiding several translating, nonrotating, nonintersecting obstacles on its way. All obstacles, the goal point, and the navigating point are confined to the interior of a star-shaped set in R/sup N/ over the time interval [0, T]. Full a priori knowledge of the goal's location and of the obstacle's trajectories is assumed. We observe that the topology of the obstacle-free space is invariant in the time interval [0, T] as long as the obstacles are nonintersecting and as long as they do not cover the goal point at any time during [0, T]. Using this fact we reduce the problem, for any fixed time t/sub 0//spl isin/[0, T], to a stationary-obstacle problem, which is then solved using the method of Rimon and Koditschek. The fact that the obstacle-free space is topologically invariant allows a solution to the moving-obstacle problem over [0, T] through a continuous deformation of the stationary-obstacle solution obtained at time t/sub 0/. We construct a vector field whose flow is in fact one such deformation. We believe that ours is the first global solution to the moving-obstacle path-planning problem which uses vector fields.

Convex-along-rays functions and star-shaped sets

We study abstract convex functions with respect to the set of all abstract affine functions of the form x→ mini=l,…,n+l[li,x] - c. We describe large classes of abstract convex functions and give conditions guaranteeing subdifferentiability of these functions. Some applications to the study of problems of global optimization are discussed.

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

Seperability of Star-Shaped Sets and its Application to an Optimization Problem

In this paper we deal with a separation problem of stat-shaped sets. It is shown that under natural assumptions two star-shaped subsets of n-dimensional Euclidean space R ncan be separated by a finite set of linear functionals. The number of this functionals is determined by the dimension of the space R n The separation theorem for star-shaped sets is used in studying of global minimum conditions

An Optimal Krasnosel'skii-type Theorem for an Open Starshaped Set

It is shown that for an open connected nonconvex set S in a real topological linear space ker\(S = \cap \) conv \(A_z :z \in \), where slnc S denotes the set of strong local nonconvexity points of S and % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa % aaleaacaWG6baabeaakiabg2da9iaacUhacaWGZbGaeyicI4Saam4u % aiaacQdacaWG6baaaa!3EFE! \[ A_z = \{ s \in S:z \] is clearly visible from s viaS . This formula, combined with familiar procedures, generates the Krasnosel'skii-type characterizations for the dimension of the kernel of S in complete separable real metric linear spaces and, provided slncS is bounded, in finite-dimensional Euclidean spaces. The latter generalizes a planar result and settles an open problem.

Noncoercive hemivariational inequality approach to constrained problems for star-shaped admissible sets

The hemivariational inequality approach is used in order to establish the existence of solutions to a large class of noncoercive constrained problems in a reflexive Banach space, in which the set of all admissible elements is not convex but fulfills some star-shaped property.

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

Star-shaped sets in normed spaces

We prove a generalization of the Krasnosel’ski theorem on star-shaped sets. Usingd-segments inn-dimensional Minkowski spaces instead of usual segments, the notions “d-visibility” and “d-star-shapedness” are introduced. Our main aim is to give necessary and sufficient conditions ford-star-shapedness in finite-dimensional normed spaces.

An extension of Brosowski-Meinardus theorem on invariant approximation

We obtain 1 generalization of a fixed point theorem of Dotson for non-expansive mappings on star-shaped sets and then use it to prove a unified Brosowski-Meinardus theorem on invariant approximation in the setting if p-normed linear spaces.

A New General Method for Constructing Confidence Sets in Arbitrary Dimensions: With Applications

Let $\mathbf{X}$ have a star unimodal distribution $P_0$ on $\mathbb{R}^p$. We describe a general method for constructing a star-shaped set $S$ with the property $P_0(\mathbf{X} \in S) \geq 1 - \alpha$, where $0 < \alpha < 1$ is fixed. This is done by using the Camp-Meidell inequality on the Minkowski functional of an arbitrary star-shaped set $S$ and then minimizing Lebesgue measure in order to obtain size-efficient sets. Conditions are obtained under which this method reproduces a level (high density) set. The general theory is then applied to two specific examples: set estimation of a multivariate normal mean using a multivariate $t$ prior and classical invariant estimation of a location vector $\mathbf{\theta}$ for a mixture model. In the Bayesian example, a number of shape properties of the posterior distribution are established in the process. These results are of independent interest as well. A computer code is available from the authors for automated application. The methods presented here permit construction of explicit confidence sets under very limited assumptions when the underlying distributions are calculationally too complex to obtain level sets.

An Optimal Krasnosel'skii-Type Theorem for the Dimension of the Kernel of a Starshaped Set

It is shown for a closed connected nonconvex and locally compact subset S of a real normed linear space that ker S = ∩ {conv Sz : z ∈ D ∩ reg S}, where reg S denotes the set of regular points of S, D is a relatively open subset of S containing the set lnc S of local nonconvexity points of S, and Sz = {s ∈ S : z is visible from s via S}. An analogous intersection formula, with the set sph S of spherical points of S in place of reg S, is shown to hold for a closed connected nonconvex set S in a real Banach space which is uniformly convex and uniformly smooth. A routine procedure then leads to a Krasnosel'skii-type characterization of the dimension of the kernel of a closed connected nonconvex subset S of Rd with lnc S bounded. This improves a recent result by removing the hypothesis of compactness of S and settles an open problem.

Ambiguous loci of the metric projection onto compact starshaped sets in a Banach space

LetE be a real Banach space andL(E) the family of all nonempty compact starshaped subsets ofE. Under the Hausdorff distance,L(E) is a complete metric space. The elements of the complement of a first Baire category subset ofL(E) are called typical elements ofL(E). ForX∈L(E) we denote byπχ the metrical projection ontoX, i.e. the mapping which associates to eacha∈E the set of all points inX closest toa. In this note we prove that, ifE is strictly convex and separable with dimE≥2, then for a typicalX∈L(E) the mapπ χ is not single valued at a dense set of points. Moreover, we show that a typical element ofL(E) has kernel consisting of one point and set of directions dense in the unit sphere ofE.

Hemivariational inequality approach to constrained problems for star-shaped admissible sets

The hemivariational inequality approach is applied to establish the existence of solutions to a large class of nonconvex constrained problems in a reflexive Banach space. The admissible sets are supposed to be star-shaped with respect to a ball. Due to a discontinuity property of the Clarke directional differential related to the corresponding distance functions, the proposed method permits one to attain the solution without passing to zero with the penalization parameter. Some applications to nonconvex constrained variational problems illustrate the theory.

Extrapolation of functions in the Denjoy class on a star-shaped compact set

Extrapolation of functions in the Denjoy class on a star-shaped compact set