Weighted covering numbers of convex sets

In this paper we define the notions of weighted covering number and weighted separation number for convex sets, and compare them to the classical covering and separation numbers. This sheds new light on the equivalence of classical covering and separation. We also provide a formula for computing these numbers via a limit of classical covering numbers in higher dimensions.     

Laws of Large Numbers for Exchangeable Random Sets in Kuratowski-Mosco Sense

Abstract

Most of the results for laws of large numbers based on Banach space valued random sets assume that the sets are independent and identically distributed (IID) and compact, in which Rådström embedding or the refined method for collection of compact and convex subsets of a Banach space plays an important role. In this paper, exchangeability among random sets as a dependency, instead of IID, is assumed in obtaining strong laws of large numbers, since some kind of dependency of random variables may be often required for many statistical analyses. Also, the Hausdorff convergence usually used is replaced by another topology, Kuratowski-Mosco convergence. Thus, we prove strong laws of large numbers for exchangeable random sets in Kuratowski-Mosco convergence, without assuming the sets are compact, which is weaker than Hausdorff sense.     

Condition numbers and error bounds in convex programming

After a brief survey on condition numbers for linear systems of equalities, we analyse error bounds for convex functions and convex sets. The canonical representation of a convex set is defined. Other representations of a convex set by a convex function are compared with the canonical representation. Then, condition numbers are introduced for convex sets and their convex representations.     

Chromatic numbers of spheres

The chromatic number of a subset of Euclidean space is the minimal number of colors sufficient for coloring all points of this subset in such a way that any two points at the distance 1 have different colors. We give new upper bounds on chromatic numbers of spheres. This also allows us to give new upper bounds on chromatic numbers of any bounded subsets.     

The Law of Large Numbers in a Metric Space with a Convex Combination Operation

We consider a separable complete metric space equipped with a convex combination operation. For such spaces, we identify the corresponding convexification operator and show that the invariant elements for this operator appear naturally as limits in the strong law of large numbers. It is shown how to uplift the suggested construction to work with subsets of the basic space in order to develop a systematic way of proving laws of large numbers for such operations with random sets.     

Almost Convex Subsets

We study almost convex subsets of spaces with one-sided curvature bounds. We derive some characterisations and properties of sets of positive reach in Riemannian manifolds.     

Quermaβintegrale zentralsymmetrischer Deckel

We give lower bounds for the volume, the surface area, and the other quermass-integrals of centro-symmetric convex universal covers in n-dimensional Euclidean spaces. The estimates are sharp in the case n = 2. The given bounds are also bounds for the quermassintegrals of convex translation covers.     

Coloring arcs of convex sets

In this note we consider Ramsey-type problems on graphs whose vertices are represented by the vertices of a convex polygon in the Euclidean plane. The edges of the graph are represented by the segments between the points of the polygon. The edges are arbitrarily colored by a fixed number of colors and the problem is to decide whether there exist monochromatic subgraphs of certain types satisfying some geometric conditions. We will give lower and upper bounds for these geometric Ramsey numbers for certain paths and cycles and also some exact values. It turns out that the particular type of the embedding is crucial for the growth rate of the corresponding geometric Ramsey numbers. In particular, the Ramsey numbers for crossing 4-cycles and t colors grow quadratically in t, while for convex 4-cycles they grow at least exponentially.     

Fuzzy real algebra: Some results

In this paper, the possibility to perform easily most of the extended n-ary operations on fuzzy subsets of the real line is shown. A general algorithm is given. These results are particularized for usual operations such as addition, subtraction, multiplication, division, ‘max’ and ‘min’ operations for normalized convex fuzzy subsets of the real line, i.e. fuzzy numbers. A three parameters representation for fuzzy numbers is shown to be very convenient to perform usual operations. Lastly, interpretative comments about fuzzy real algebra are given and possible applications pointed out.     

Mean square inequalities for chords of convex sets

This paper is concerned with establishing lower bounds for the integrals of the square of the lengths of area and perimeter bisecting chords of planar convex sets. The results obtained provide verification of two recent conjectures of Lutwak. When combined with the known upper bounds for these integrals they yield the classical isoperimetric inequality. The main proof technique involves estimation of the winding numbers of the locus of the midpoints of the chords concerned.     

A Brunn-Minkowski inequality for the integer lattice

A close discrete analog of the classical Brunn-Minkowksi inequality that holds for finite subsets of the integer lattice is obtained. This is applied to obtain strong new lower bounds for the cardinality of the sum of two finite sets, one of which has full dimension, and, in fact, a method for computing the exact lower bound in this situation, given the dimension of the lattice and the cardinalities of the two sets. These bounds in turn imply corresponding new bounds for the lattice point enumerator of the Minkowski sum of two convex lattice polytopes. A Rogers-Shephard type inequality for the lattice point enumerator in the plane is also proved.

     

Power-law estimates for the central limit theorem for convex sets

We investigate the rate of convergence in the central limit theorem for convex sets established in [B. Klartag, A central limit theorem for convex sets, Invent. Math., in press. [8]]. We obtain bounds with a power-law dependence on the dimension. These bounds are asymptotically better than the logarithmic estimates which follow from the original proof of the central limit theorem for convex sets.     

Average equidistribution and statistical independence properties of digital inversive pseudorandom numbers over parts of the period

This article deals with the digital inversive method for generating uniform pseudorandom numbers. Equidistribution and statistical independence properties of the generated pseudorandom number sequences over parts of the period are studied based on the distribution of tuples of successive terms in the sequence. The main result is an upper bound for the average value of the star discrepancy of the corresponding point sets. Additionally, lower bounds for the star discrepancy are established. The method of proof relies on bounds for exponential sums.

     

Almost Chebyshev set with respect to bounded subsets

The uniqueness and existence of restricted Chebyshev center with respect to arbitrary subset are investigated. The concept of almost Chebyshev sets with respect to bounded subsets is introduced. It is proved that each closed subset in a reflexive locally uniformly convex (uniformly convex, respectively) Banach space is an almost Chebyshev subset with respect to compact convex subsets (bounded convex subsets and bounded subsets, respectively). Project supported by the National Natural Science Foundation of China, Natural Science Foundation of Zhejiang Province, and the State Major Key Project for Basic Researchers of China.     

On the Chromatic Number of Subsets of the Euclidean Plane

The chromatic number of a subset of the real plane is the smallest number of colors assigned to the elements of that set such that no two points at distance 1 receive the same color. It is known that the chromatic number of the plane is at least 4 and at most 7. In this note, we determine the bounds on the chromatic number for several classes of subsets of the plane such as extensions of the rational plane, sets in convex position, infinite strips, and parallel lines.     

Entropy extension

We prove an “entropy extension-lifting theorem.” It consists of two inequalities for the covering numbers of two symmetric convex bodies. The first inequality, which can be called an “entropy extension theorem,” provides estimates in terms of entropy of sections and should be compared with the extension property of ?_∞. The second one, which can be called an “entropy lifting theorem,” provides estimates in terms of entropies of projections.     

Subadditive ergodic theorems for random sets in infinite dimensions

We prove pointwise and mean versions of the subadditive ergodic theorem for superstationary families of compact, convex random subsets of a real Banach space, extending previously known results that were obtained in finite dimensions or with additional hypotheses on the random sets. We also show how the techniques can be used to obtain the strong law of large numbers for pairwise independent random sets, as well as results in the weak topology.     

Average equidistribution properties of compound nonlinear congruential pseudorandom numbers

The present paper deals with the compound nonlinear congruential method for generating uniform pseudorandom numbers, which has been introduced recently. Equidistribution properties of the generated sequences over parts of the period are studied, based on the discrepancy of the corresponding point sets. Upper and lower bounds for the average value of these discrepancies are established, which are essentially best possible. These results show that the average equidistribution behavior of compound nonlinear congruential pseudorandom numbers fits well the equidistribution properties of true random numbers. The method of proof relies heavily on estimates of the average value of incomplete exponential sums.

     

A class of bounds for convex bodies in Hilbert space

We extend to infinite dimensions a class of bounds forL _p metrics of finite-dimensional convex bodies. A generalization to arbitrary increasing convex functions is done simultaneously. The main tool is the use of Gaussian measure to effect a normalization for varying dimension. At a point in the proof we also invoke a strong law of large numbers for random sets to produce a rotational averaging.Supported in part by ONR Grant N0014-90-J-1641 and NSF Grant DMS-9002665.     

Reduction of quasidifferentials and minimal representations

Some criterias for the non-minimality of pairs of compact convex sets of a real locally convex topological vector space are proved, based on a reduction technique via cutting planes and excision of compact convex subsets. Following an example of J. Grzybowski, we construct a class of equivalent minimal pairs of compact convex sets which are not connected by translations.Corresponding author.