Increasing unions of starlike sets

Let S be an arbitrary set in R². Every three points of cl S are clearly visible via S from a common point of cl S if and only if each bounded subset of S may be extended to a starlike set in S. When this occurs, set S is expressible as an increasing union of starlike sets.     

Families of planar sets having starlike union

Let be a finite family of compact sets in the plane, and letk be a fixed natural number. If every three (not necessarily distinct) members of have a union which is simply connected and starshaped viak-paths, then and is starshaped viak-paths. Analogous results hold for paths of length at most , > 0, and for staircase paths, although not for staircasek-paths.Supported in part by NSF grant DMS-9504249     

Clear visibility,starshaped sets,and finitely starlike sets

Let S be an arbitrary nonempty set in R^d. The following results are true for every k, 0kd: the dimension of ker S is at least k if and only if every countable family of boundary points of S is clearly visible from a common k-dimensional neighborhood in S. Similarly, ker S contains a k-dimensional -neighborhood if and only if every countable family of boundary points of S is clearly visible from a common k-dimensional -neighborhood in S.In the plane, we have the following results concerning finitely starlike sets: for S an arbitrary nonempty set in R², S is finitely starlike if every three points of cl S are clearly visible from a common point of S. In case S –R² and int cl SS=, then S is finitely starlike if and only if every three points of S are visible from a common point of S. In each case, the number 3 is best possible.     

On paraconvexity of graphs of continuous functions

The concept of paraconvexity of a subsetP E of a normed spaceE was first introduced by E. Michael. Roughly speaking, it consists of a controlled weakening of the convexity assumption forP, where the control is guaranteed via some parameter [0, 1). In this paper, we consider the case whenP is a subset of some (n+1)-dimensional Euclidean spaceE andP is the graph of some continuous functionf:V , whereV E is some convexn-dimensional subset ofE. Our key result is that paraconvexity of such a setP follows from the paraconvexity of sections ofP by two-dimensional planes, orthogonal toV. As an application, we prove a selection theorem for graph-valued mappings whose values have Lipschitzian (with a fixed constant) or monotone two-dimensional sections.Supported in part by the Ministry of Science and Technology of the Republic of Slovenia Research Grant No. P1-0214-101-93.Supported in part by G. Soros International Science Foundation.     

Finitely starlike sets whose F-stars have positive measure

Let and assume that there is a countable collection of lines {L _i : 1 i} such that (int cl S) and ((int cl S) S) L _i has one-dimensional Lebesgue measure zero, 1 i. Then every 4 point subset ofS sees viaS a set of positive two-dimensional Lebesgue measure if and only if every finite subset ofS sees viaS such a set. Furthermore, a parallel result holds with two-dimensional replaced by one-dimensional. Finally, setS is finitely starlike if and only if every 5 points ofS see viaS a common point. In each case, the number 4 or 5 is best possible.Supported in part by NSF grant DMS-8705336.     

Krasnosel'skii numbers for bounded finitely starlike sets inR d

For eachk andd, 1kd, definef(d, d)=d+1 andf(d, k)=2d if 1kd–1. The following results are established:Let be a uniformly bounded collection of compact, convex sets inR ^d . For a fixedk, 1kd, dim {MM in }k if and only if for some > 0, everyf(d, k) members of contain a commonk-dimensional set of measure (volume) at least.LetS be a bounded subset ofR ^d . Assume that for some fixedk, 1kd, there exists a countable family of (k–l)-flats {H _i :i1} inR ^d such that clS S {H_i i 1 } and for eachi1, (clS S) H _i has (k–1) dimensional measure zero. Every finite subset ofS sees viaS a set of positivek-dimensional measure if and only if for some>0, everyf(d,k) points ofS see viaS a set ofk-dimensional measure at least .The numbers off(d,d) andf(d, 1) above are best possible.Supported in part by NSF grant DMS-8705336.     

Some Krasnosel'skii numbers for finitely starlike sets in the plane

Let S be a subset of the plane. In case (int cl S) S = , then S is finitely starlike if and only if every 4 points of S see via S a common point. In case (int cl S) S has at most countably many components, each a singleton set, then S is finitely starlike if and only if every 5 points of S see via S a common point. Each of the numbers 4 and 5 is best possible. Examples show that these results fail without suitable restrictions on (int cl S) S. Moreover, a final example shows that if a general Krasnosel'skii number . exists to characterize finitely starlike sets in the plane, then > 9.     

On rotationally starlike logharmonic mappings

This paper considers the class of all mappings of the form where h and g are analytic in the unit disk U, normalized by , and such that is logharmonic with respect to an analytic self‐map a of U. A distortion estimate and the radius of starlikeness are obtained for this class. Additionally, a solution to the problem of minimizing the moments of order p over the class is found, as well as an estimate for arclength.     

On weakly starlike multivalent functions

Journal d'Analyse Mathématique -     

On a generalization of starlike functions

In this paper, we mainly study the order of $q$-starlikeness of the well-known basic hypergeometric function. In addition, we discuss the Bieberbach-type problem and the second order Hankel determinant for a generalized class of starlike functions.     

A Krasnosel'skii theorem for open finitely starlike sets in R 3

Let S be an open set in R ³ such that: (1) whenever three points x, y, z in S see each other via S, then conv{x, y, z} S, and (2) every seven points in S see via S a common point. Then S is finitely starlike. The proof uses the topological version of Helly's theorem.     

On set-valued mappings with starlike graphs

The paper studies some topological properties of starlike bodies. It is proved that the boundary of a starlike body is a Lipschitz surface. A separability theorem for starlike bodies is proved. It is shown that under some additional assumptions the starlike property of the graph provides the local Lipschitz property of the set-valued mapping itself. It is shown that F. Clark’s contingent and tangential cones are Boltyansky tents. On the base of these results, some lower and upper differentials for set-valued mappings with starlike graphs are constructed. Some theorems on fixed points of set-valued mappings with starlike values are proved.     

On Ramsey numbers involving starlike multipartite graphs

The Ramsey number r(G, H) is evaluated exactly in certain cases in which both G and H are complete multipartite graphs K(n,₁, n₂, …. n_k). Specifically, each of the following cases is handled whenever n is sufficiently large: r(K(1, m₁, …. m_k), K(1, n)), r(K(1, m), K(n₁, …. n_k, n)), provided m ≧ 4, and r(K(1, 1, m), K(n_k, …, n_k, n)).     

Norm estimates and representations for Calderón-Zygmund operators using averages over starlike sets

We show that homogeneous singular integrals may be represented in terms of averages over starlike sets. This permits us to use the geometry of starlike sets to derive operator-specific weighted norm inequalities.

     

Local paraconvexity and local selection theorems

Theorems on the local extendability of selections for non-convex-valued maps of paracompact spaces into Banach spaces, i.e., infinite-dimensional analogs of the finite-dimensional Michael selection theorem are proved. We were able to obtain these results under an appropriate metric control of the local degree of nonconvexity on the valuesF(x), which naturally leads us to introduce the notion of equi-locally paraconvex families of sets. It is shown that all convex subsets of the integral curves of the differential equationy′=f(x,y) with a continuous right-hand sidef and the isometric images of such subsets form an equi-locally paraconvex family of subsets of a Euclidean space. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 261–269, February, 1999.     

On the eigenvalues and spectral radius of starlike trees

Let \(k\ge 1\) and \(n_1,\ldots ,n_k\ge 1\) be some integers. Let \(S(n_1,\ldots ,n_k)\) be a tree T such that T has a vertex v of degree k and \(T{\setminus } v\) is the disjoint union of the paths \(P_{n_1},\ldots ,P_{n_k}\), that is \(T{\setminus } v\cong P_{n_1}\cup \cdots \cup P_{n_k}\) so that every neighbor of v in T has degree one or two. The tree \(S(n_1,\ldots ,n_k)\) is called starlike tree, a tree with exactly one vertex of degree greater than two, if \(k\ge 3\). In this paper we obtain the eigenvalues of starlike trees. We find some bounds for the largest eigenvalue (for the spectral radius) of starlike trees. In particular we prove that if \(k\ge 4\) and \(n_1,\ldots ,n_k\ge 2\), then \(\frac{k-1}{\sqrt{k-2}}<\lambda _1(S(n_1,\ldots ,n_k))<\frac{k}{\sqrt{k-1}}\), where \(\lambda _1(T)\) is the largest eigenvalue of T. Finally we characterize all starlike trees that all of whose eigenvalues are in the interval \((-2,2)\).