From a (p, 2)-Theorem to a Tight (p, q)-Theorem

A Helly-type theorem for higher-dimensional transversals

Let X be a separable Banach space, G a LCA group, and let L1(G, P KC (X))be the space of all compact convex set-valued functions defined on a locally compact abelian group G, where P KC (X) denotes the family of compact convex sets of X. L1 is a metric space under the metric for any F1, F2 ϵ L1 (G,P KC (X))where h denotes the Hausdorff metric on P KC (X). We formulate the convolution of Ll and L1(G), and prove that any set-valued mapping F ϵ L1 (G,P KC (X)) is approximated by the convolution of an approximate identity {e a } for L1(G) and F self under a specific metric topology D1 in L1. This property will be extended to study Fourier analysis of set-valued maps.

LetK be a family of compact convex sets in the plane. We show that if every three members ofK admit a common line transversal, then there exist four lines which together meet all the members ofK.

OF THE DISSERTATION Random matrices and random boxes by Linh V. Tran Dissertation Director: Prof. Van Vu This thesis concerns two questions on random structures: the semi-circular law for adjacency matrix of regular random graph and the piercing number for random boxes. • Random matrices We proved in full generality the semi-circular law for random d-regular graph model in the case d tends to infinity as n does. Our result complements the McKay law [19], which applied for the case d is an absolute constant. • Random boxes. Take n random boxes with axis-parallel edges inside the unit cube [0, 1]d, the piercing number τd(n) is the minimum number of points needed to pierce all boxes. Using hypergraph setting, we was able to prove a near sharp estimation for τd(n): for any d ≥ 2 Ωd( √ n(log n)d/2−1) = τd(n) = Od( √ n(log n)d/2−1 log log n). This thesis is based on two papers by the author [31] and [30] (joint work with Van Vu and Ke Wang).

Intersection patterns of convex sets in ℝ^d have the remarkable property that for d+1 ≤ k ≤ 𝓁, in any sufficiently large family of convex sets in ℝ^d, if a constant fraction of the k-element subfamilies have nonempty intersection, then a constant fraction of the 𝓁-element subfamilies must also have nonempty intersection. Here, we prove that a similar phenomenon holds for any topological set system ℱ in ℝ^d. Quantitatively, our bounds depend on how complicated the intersection of 𝓁 elements of ℱ can be, as measured by the maximum of the ⌈d/2⌉ first Betti numbers. As an application, we improve the fractional Helly number of set systems with bounded topological complexity due to the third author, from a Ramsey number down to d+1. We also shed some light on a conjecture of Kalai and Meshulam on intersection patterns of sets with bounded homological VC dimension. A key ingredient in our proof is the use of the stair convexity of Bukh, Matousek and Nivasch to recast a simplicial complex as a homological minor of a cubical complex.

On the equivalence of extrapolation and Richardson’s iteration and its applications

k-Dimensional intersections of convex sets and convex kernels

Article A Helly-type theorem for hyperplane transversals to well-separated convex sets Share on Authors: Boris Aronov Polytechnic University, Brooklyn, NY Polytechnic University, Brooklyn, NYView Profile , Jacob E. Goodman City College, City University of New York, New York, NY City College, City University of New York, New York, NYView Profile , Richard Pollack Courant Institute of Mathematical Sciences, New York University, New York, NY Courant Institute of Mathematical Sciences, New York University, New York, NYView Profile , Rephael Wenger The Ohio State University, Columbus, OH The Ohio State University, Columbus, OHView Profile Authors Info & Claims SCG '00: Proceedings of the sixteenth annual symposium on Computational geometryMay 2000 Pages 57–63https://doi.org/10.1145/336154.336178Online:01 May 2000Publication History 0citation233DownloadsMetricsTotal Citations0Total Downloads233Last 12 Months4Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

The aim of the present paper is to investigate some convexities. First we introduce and investigate the "pseudoconvex" spaces and give a minimax theorem applying the ideas of [8]. The notion of convex spaces was introduced by Komiya [1] (see below for details). In [2] the authors proved a Nikaidd--Isoda-type theorem for convex spaces. An interesting convexity notion, not giving a convex space in the sense of [1] is obtained in [5]. The paper [4] contains investigations with respect to this convexity structure, analogous to that of in [2]. Now we give a common generalization of the convexities of [i] and [5], the so-called pseudoconvex space. It turns out that the compact pseudoconvex spaces have the fixed point property and a Nikaido-Isoda-type theorem holds. First we prove these assertions and after we prove an inequality between the Helly and Caratheodory number of pseudoconvex spaces. In connection with these investigations we mention the work of M. Horv~th [3] where the Helly, Caratheodory and Radon numbers of a special convexity structure are calculated and whose proof is based on graph theoretical results. At last we continue the investigation of V. Komornik [10] and give a related minimax theorem in interval space.

Let S be a finite collection of compact convex sets in \R d . Let D(S) be the largest diameter of any member of S . We say that the collection S is ?-separated if, for every 0 < k < d , any k of the sets can be separated from any other d-k of the sets by a hyperplane more than ? D(S)/2 away from all d of the sets. We prove that if S is an ? -separated collection of at least N(?) compact convex sets in \R d and every 2d+2 members of S are met by a hyperplane, then there is a hyperplane meeting all the members of S . The number N(?) depends both on the dimension d and on the separation parameter ? . This is the first Helly-type theorem known for hyperplane transversals to compact convex sets of arbitrary shape in dimension greater than one.

It is proved that if the nonempty intersection of bounded closed convex sets A \ B is contained in (A u F) ( (B u F) and one of the following holds true: (i) the space X is less-than-three dimensional, (ii) A ( B is convex, (iii) F is a one-point set, then A \ B A u F or A \ B B u F (Theorems 2 and 3). Moreover, under some hypotheses the characterization of A and B such that A \ B is a summand of A ( B is given (Theorem 3). In this paper we consider the inclusion A\B (AuF)((B uF) which plays a central role in proving criterion of convex minimality of pairs of bounded closed convex sets (12), (13). Minimal pairs of compact convex sets were investigated in a series of papers (2), (4), (5), (10), (12), and others. Pairs of compact convex sets naturally arise in quasidierential

Hyperspaces of convex bodies of constant width

It is shown that the problem of the best uniform approximation in the Hausdorff metric of a continuous set-valued map with finite-dimensional compact convex images by constant set-valued maps whose images are balls in some norm can be reduced to a visual geometric problem. The latter consists in constructing a spherical layer of minimal thickness which contains the complement of a compact convex set to a larger compact convex set.

Compact convex sets with a special property (P-property) in a finite-dimensional space are considered. This property ensures a certain regularity of the boundaries of such sets. A fairly complete description of the class of compact convex sets with the P-property is given. It is shown that this class is closed with respect to the basic set operations. The application of the P-property is exemplified; in particular, it is shown that any generating set has the P-property.

We study the convergence of an iterative projection/reflection algorithm originally proposed for solving what are known as phase retrieval problems in optics. There are two features that frustrate any analysis of iterative methods for solving the phase retrieval problem: nonconvexity and infeasibility. The algorithm that we developed, called relaxed averaged alternating reflections (RAAR), was designed primarily to address infeasibility, though our strategy has advantages for nonconvex problems as well. In the present work we investigate the asymptotic behavior of the RAAR algorithm for the general problem of finding points that achieve the minimum distance between two closed convex sets in a Hilbert space with empty intersection, and for the problem of finding points that achieve a local minimum distance between one closed convex set and a closed prox-regular set, also possibly nonintersecting. The nonconvex theory includes and expands prior results limited to convex sets with nonempty intersection. To place the RAAR algorithm in context, we develop parallel statements about the standard alternating projections algorithm and gradient descent. All of the various algorithms are unified as instances of iterated averaged alternating proximal reflectors applied to a sum of regularized maximal monotone mappings.