Lower bounds to helly numbers of line transversals to disjoint congruent balls

A line ℓ is a transversal to a family F of convex objects in ℝ ^d if it intersects every member of F. In this paper we show that for every integer d ⩾ 3 there exists a family of 2d−1 pairwise disjoint unit balls in ℝ ^d with the property that every subfamily of size 2d − 2 admits a transversal, yet any line misses at least one member of the family. This answers a question of Danzer from 1957. Crucial to the proof is the notion of a pinned transversal, which means an isolated point in the space of transversals. Here we investigate minimal pinning configurations and construct a family F of 2d−1 disjoint unit balls in ℝ ^d with the following properties: (i) The space of transversals to F is a single point and (ii) the space of transversals to any proper subfamily of F is a connected set with non-empty interior.     

Networks and closed balls

Neural networks calledtangent networks are constructed by explicit reference to the geometry of a set, and then blended intocascades which approximate characteristic functions of closed balls. In this way some known results about approximation by single hidden layer neural networks are re-proved in a very constructive and geometrical fashion.     

Convex sets which are intersections of closed balls

In this paper we aim to investigate different questions concerning the stability of the set of all intersections of closed balls in a normed space. We are mainly concerned with: (i) the stability of under the closure of the vector sums; (ii) the stability under the addition of balls. We prove that (i) and (ii) are different properties which have strong connections with the geometry of the space. They have interest both in finite and infinite dimension. In the former case, there is a link with linear programming theory. We also study two more stability properties related to the well-known binary intersection property. Mazur sets and Mazur spaces are introduced, as a natural family satisfying (i). We prove that every two-dimensional normed space is a Mazur space, a result which distinguishes dimension d?2 from dimension d?3. We also discuss the connections between Mazur spaces and porosity.     

Higher order secant varieties and blown up varieties

Let π:X→Y be the blowing up of the projective varietyY at s general points. Here we study the higher order secant varieties of the linearly normal embeddings ofX andY into projective spaces. We give conditions on the embedding ofY which imply that the firstt secant varieties of a related embedding ofX have the expected dimension.     

Closed transversals and the genus of closed leaves in foliated 3-manifolds

The classical local theory of integrable 2-plane fields in 3-space leads to interesting qualitative questions about the global properties of solutions surface (i.e., leaves of a foliation) on 3-manifolds. It is now known that foliations admitting a closed leaf of suitably high genus abound on all closed or orientable 3-manifolds that are not rational homology spheres (S. Goodman, Proc. Nat. Acad. Sci. U.S.A.71 (1974), 4414–4415), and this leads to natural questions about the “positions” of such leaves relative to the rest of the foliation. One such question, suggested by Goodman's theorem on closed transversals (S. Goodman, ibid.), is considered here.     

Higher order secant varieties and blown up varieties

Let π:X→Y be the blowing up of the projective varietyY at s general points. Here we study the higher order secant varieties of the linearly normal embeddings ofX andY into projective spaces. We give conditions on the embedding ofY which imply that the firstt secant varieties of a related embedding ofX have the expected dimension.     

Projective embeddings of projective schemes blown up at subschemes

Suppose X is a nonsingular projective scheme, Z a nonsingular closed subscheme of X. Let X be the blowup of X centered at Z, E ₀ the pull-back of a general hyperplane in X, and E the exceptional divisor. In this paper, we study projective embeddings of X given by divisors . When X satisfies a necessary condition, we give explicit values of d and such that for all e>0 and embeds X as a projectively normal and arithmetically Cohen-Macaulay scheme. We also give a uniform bound for the regularities of the ideal sheaves of these embeddings, and study their asymptotic behaviour as t gets large compared to e. When X is a surface and Z is a 0-dimensional subscheme, we further show that these embeddings possess property N _p for all te>0. Mathematics Subject Classification (2000):14E25, 14M05, 13H10.Dedicated to the sixtieth birthday of Prof. A.V. Geramita     

Slope instability of projective spaces blown up along a line

Let \(\text {Bl}_{\mathbb {P}^1} \mathbb {P}^n\) be a Kähler manifold obtained by blowing up a complex projective space \(\mathbb {P}^n\) along a line \(\mathbb {P}^1\). We prove that \(\text {Bl}_{\mathbb {P}^1} \mathbb {P}^n\) is slope unstable with respect to any polarisation, and hence, it does not admit constant scalar curvature Kähler metrics in any rational Kähler class.     

Blow up of balls and coverings in metric spaces

We obtain some refinements and extensions of the Basic Covering Theorem in a metric space (X, ρ). The properties of the metric ρ are used to define an inclusion coefficient k in this theorem, and this is related to the supremum of numbers t such that ρ ^t is a metric in X. The inclusion coefficient k characterizes ultrametric spaces.     

Point transversals to translates of a trapezoid

Anm-transversal to a family of convex sets in the plane is anm-point set which intersects every members of the family. One of Grübaum’s conjectures says that a planar family of translates of a convex compact set has a 3-transversal provided that any two of its members intersect. Recently the conjecture has been proved affirmatively (see [4]). In the present paper we provide a different and straightforward proof for the conjecture for the family of translates of a closed trapezoid in the plane and give several concrete 3-transversals.     

Topological transversals to a family of convex sets

Let Fmathcal{F} be a family of compact convex sets in ℝ ^d . We say that Fmathcal{F} has a topological ρ-transversal of index (m,k) (ρ<m, 0<k≤d−m) if there are, homologically, as many transversal m-planes to Fmathcal{F} as m-planes containing a fixed ρ-plane in ℝ ^m+k .     

Large transversals to small families of unit disks

Summary It is proved that if the nonempty intersection of bounded closed convex sets AnB is contained in (A + F)U(B+F) and one of the following holds true: (i) the space X is less-than-three dimensional, (ii) AUB is convex, (iii) F is a one-point set, then AnBCA+F or AnBCB+F (Theorems 2 and 3). Moreover, under some hypotheses the characterization of A and B such that AnB is a summand of AUB is given (Theorem 3).     

Groups generating transversals to semisimple lie group actions

We describe those discrete groups with finite measure preserving actions that are stably orbit equivalent to such an action of a higher rank simple Lie group. This is applied to obtain information on the question of when ergodic equivalence relations are generated by a free action of a group. Research partially supported by the National Science Foundation and the Israel-U.S. Binational Science Foundation.     

Relatively weakly open sets in closed balls of Banach spaces,and the centralizer

Let f be a transcendental entire function and let I(f) denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, I(f) is connected. In particular, we show that I(f) is connected if f has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Eremenko’s conjecture that I(f) has no bounded components is true. We also give a new criterion related to I(f) which is sufficient to ensure that f has no unbounded Fatou components.     

From approximate balls to approximate ellipses

A ball spans a set of n points when none of the points lie outside it. In Zarrabi-Zadeh and Chan (Proceedings of the 18th Canadian conference on computational geometry (CCCG’06), pp 139–142, 2006) proposed an algorithm to compute an approximate spanning ball in the streaming model of computation, and showed that the radius of the approximate ball is within 3/2 of the minimum. Spurred by this, in this paper we consider the 2-dimensional extension of this result: computation of spanning ellipses. The ball algorithm is simple to the point of being trivial, but the extension of the algorithm to ellipses is non-trivial. Surprisingly, the area of the approximate ellipse computed by this approach is not within a constant factor of the minimum and we provide an elegant proof of this. We have implemented this algorithm, and experiments with a variety of inputs, except for a very pathological one, show that it can nevertheless serve as a good heuristic for computing an approximate ellipse.     

Flat transversals to flats and convex sets of a fixed dimension

Helly and Hadwiger type theorems for transversal m-flats to families of flats and, respectively, convex sets of dimension n are proved in the case of general position. The proofs rely on Helly type theorems for “linear partitions” and “convex partitions,” so that a general theory of Helly numbers is also developed.