Periodic Billiard Trajectories in Smooth Convex Bodies

We consider billiard trajectories in a smooth convex body in \mathbbR^d{\mathbb{R}^{d}} and estimate the number of distinct periodic trajectories that make exactly p reflections per period at the boundary of the body. In the case of prime p we obtain the lower bound (d – 2)(p – 1) + 2, which is much better than the previous estimates.     

Lower estimates for the number of closed trajectories of generalized billiards

The mathematical study of periodic billiard trajectories is a classical question that goes back to George Birkhoff. A billiard is the motion of a particle in the absence of field of force. Trajectories of such a particle are geodesics. A billiard ball rebounds from the boundary of a given domain making the angle of incidence equal the angle of reflection. Let k be a fixed integer. Birkhoff proved a lower estimate for the number of closed billiard trajectories of length k in an arbitrary plane domain. We give a general definition of a closed billiard trajectory when the billiard ball rebounds from a submanifold of a Euclidean space. Besides, we show how in this case one can apply the Morse inequalities using the natural symmetry (a closed polygon may be considered starting at any of its vertices and with the reversed direction). Finally, we prove the following estimate. Let M be a smooth closed m-dimensional submanifold of a Euclidean space, and let p > 2 be a prime integer. Then M has at least

Independence Numbers of Planar Contact Graphs

Abstract. We show that for a large class of convex discs C (including strictly convex discs), there exists an ε=ε(C)>0 such that the independence number of the contact graph of any packing of n translates of C in the plane is at least (1/4 + ε)n . For C a circle, we improve the lower bound of Csizmadia to 8/31n .     

Independence Numbers of Planar Contact Graphs

   Abstract. We show that for a large class of convex discs C (including strictly convex discs), there exists an ε=ε(C)>0 such that the independence number of the contact graph of any packing of n translates of C in the plane is at least (1/4 + ε)n . For C a circle, we improve the lower bound of Csizmadia to 8/31n .     

<Emphasis Type="Italic">ε</Emphasis>-Optimality and <Emphasis Type="Italic">ε</Emphasis>-Lagrangian Duality for a Nonconvex Programming Problem with an Infinite Number of Constraints

In this paper, ε-optimality conditions are given for a nonconvex programming problem which has an infinite number of constraints. The objective function and the constraint functions are supposed to be locally Lipschitz on a Banach space. In a first part, we introduce the concept of regular ε-solution and propose a generalization of the Karush-Kuhn-Tucker conditions. These conditions are up to ε and are obtained by weakening the classical complementarity conditions. Furthermore, they are satisfied without assuming any constraint qualification. Then, we prove that these conditions are also sufficient for ε-optimality when the constraints are convex and the objective function is ε-semiconvex. In a second part, we define quasisaddlepoints associated with an ε-Lagrangian functional and we investigate their relationships with the generalized KKT conditions. In particular, we formulate a Wolfe-type dual problem which allows us to present ε-duality theorems and relationships between the KKT conditions and regular ε-solutions for the dual. Finally, we apply these results to two important infinite programming problems: the cone-constrained convex problem and the semidefinite programming problem.     

Points surrounding the origin

Suppose d > 2, n > d+1, and we have a set P of n points in d-dimensional Euclidean space. Then P contains a subset Q of d points such that for any p ∈ P, the convex hull of Q∪{p} does not contain the origin in its interior. We also show that for non-empty, finite point sets A ₁, ..., A _d+1 in ℝ ^d , if the origin is contained in the convex hull of A _i ∪ A _j for all 1≤i<j≤d+1, then there is a simplex S containing the origin such that |S∩A _i |=1 for every 1≤i≤d+1. This is a generalization of Bárány’s colored Carathéodory theorem, and in a dual version, it gives a spherical version of Lovász’ colored Helly theorem. Dedicated to Imre Bárány, Gábor Fejes Tóth, László Lovász, and Endre Makai on the occasion of their sixtieth birthdays. Supported by the Norwegian research council project number: 166618, and BK 21 Project, KAIST. Part of the research was conducted while visiting the Courant Institute of Mathematical Sciences. Supported by NSF Grant CCF-05-14079, and by grants from NSA, PSC-CUNY, the Hungarian Research Foundation OTKA, and BSF.     

A Dynamical System Associated with Newton's Method for Parametric Approximations of Convex Minimization Problems

We study the existence and asymptotic convergence when t→+∞ for the trajectories generated by where is a parametric family of convex functions which approximates a given convex function f we want to minimize, and ε(t) is a parametrization such that ε(t)→ 0 when t→+∞ . This method is obtained from the following variational characterization of Newton's method: where H is a real Hilbert space. We find conditions on the approximating family and the parametrization to ensure the norm convergence of the solution trajectories u(t) toward a particular minimizer of f . The asymptotic estimates obtained allow us to study the rate of convergence as well. The results are illustrated through some applications to barrier and penalty methods for linear programming, and to viscosity methods for an abstract noncoercive variational problem. Comparisons with the steepest descent method are also provided. Accepted 5 December 1996     

A Lower Bound for Weak ɛ -Nets in High Dimension

   Abstract. A finite set N ⊂ R ^d is a weak ε-net for an n -point set X ⊂ R ^d (with respect to convex sets) if it intersects each convex set K with |K ∩ X| ≥ ε n . It is shown that there are point sets X ⊂ R ^d for which every weak ε -net has at least const ⋅

Cutting hyperplane arrangements

We consider a collectionH ofn hyperplanes in E ^d (where the dimensiond is fixed). An ε-cutting forH is a collection of (possibly unbounded)d-dimensional simplices with disjoint interors, which cover all E ^d and such that the interior of any simplex is intersected by at mostεn hyperplanes ofH. We give a deterministic algorithm for finding a (1/r)-cutting withO(r ^d ) simplices (which is asymptotically optimal). Forr≤n ^1−σ, where δ>0 is arbitrary but fixed, the running time of this algorithm isO(n(logn) ^O(1) r ^d−1). In the plane we achieve a time boundO(nr) forr≤n ^1−δ, which is optimal if we also want to compute the collection of lines intersecting each simplex of the cutting. This improves a result of Agarwal, and gives a conceptually simpler algorithm. For ann point setX⊆E ^d and a parameterr, we can deterministically compute a (1/r)-net of sizeO(rlogr) for the range space (X, {X ϒ R; R is a simplex}), In timeO(n(logn) ^O(1) r ^d−1 +r ^O(1)). The size of the (1/r)-net matches the best known existence result. By a simple transformation, this allows us to find ε-nets for other range spaces usually encountered in computational geometry. These results have numerous applications for derandomizing algorithms in computational geometry without affecting their running time significantly. A preliminary version of this paper appeared inProceedings of the Sixth ACM Symposium on Computational Geometry, Berkeley, 1990, pp. 1–9. Work on this paper was supported by DIMACS Center.     

Exponential decay rate of survival probability in a disastrous random environment

Summary. We study the exponential decay rate of the survival probability up to time t>0 of a random walker moving in Zopf; ^d in a temporally and spatially fluctuating random environment. When the random walker has a speed parameter κ>0, we investigate the influence of κ on the exponential decay rate λ(d,κ). In particular we prove that for any fixed d≥1, λ(d,κ) behaves like as logκ as κ↘0. Received: 21 May 1996 / In revised form: 2 February 1997     

On the π-function for nonconvex <Emphasis Type="Italic">l</Emphasis><Subscript>2,<Emphasis Type="Italic">p</Emphasis></Subscript>-circle discs

For 0 < p < 1, circle numbers π(p) are defined to reflect the Euclidean area-content property A _p(r) = π(p)r ² and circumference property {ie332-01} of the l _2,p -circle discs with p-generalized radius r, where the arc-length measure {ie332-02} is based upon the nonconvex star-shaped set {ie332-03} with p** > 0 satisfying {ie332-04}. The resulting π-function extends the function p → π(p) recently defined in [2] from the case of convex discs, p ⩾ 1, to the nonconvex case 0 < p < 1. This function is continuous, increasing, and takes values in (0, 2). The presented approach can be considered as reflecting a modified method of indivisibles in the sense that the indivisibles are the l _2,p -circles and that integrating their S(p**)-arc-lengths is equivalent to measuring the Euclidean area content.     

Analysis of Incomplete Data and an Intrinsic-Dimension Helly Theorem

The analysis of incomplete data is a long-standing challenge in practical statistics. When, as is typical, data objects are represented by points in ℝ ^d , incomplete data objects correspond to affine subspaces (lines or Δ-flats). With this motivation we study the problem of finding the minimum intersection radius r(ℒ) of a set of lines or Δ-flats ℒ: the least r such that there is a ball of radius r intersecting every flat in ℒ. Known algorithms for finding the minimum enclosing ball for a point set (or clustering by several balls) do not easily extend to higher-dimensional flats, primarily because “distances” between flats do not satisfy the triangle inequality. In this paper we show how to restore geometry (i.e., a substitute for the triangle inequality) to the problem, through a new analog of Helly’s theorem. This “intrinsic-dimension” Helly theorem states: for any family ℒ of Δ-dimensional convex sets in a Hilbert space, there exist Δ+2 sets ℒ′⊆ℒ such that r(ℒ)≤2r(ℒ′). Based upon this we present an algorithm that computes a (1+ε)-core set ℒ′⊆ℒ, |ℒ′|=O(Δ ⁴/ε), such that the ball centered at a point c with radius (1+ε)r(ℒ′) intersects every element of ℒ. The running time of the algorithm is O(n ^Δ+1 dpoly (Δ/ε)). For the case of lines or line segments (Δ=1), the (expected) running time of the algorithm can be improved to O(ndpoly (1/ε)). We note that the size of the core set depends only on the dimension of the input objects and is independent of the input size n and the dimension d of the ambient space. An extended abstract appeared in ACM–SIAM Symposium on Discrete Algorithms, 2006. Work was done when J. Gao was with Center for the Mathematics of Information, California Institute of Technology. Work was done when M. Langberg was a postdoctoral scholar at the California Institute of Technology. Research supported in part by NSF grant CCF-0346991. Research of L.J. Schulman supported in part by an NSF ITR and the Okawa Foundation.     

Boundary Behavior of Harmonic Functions for Truncated Stable Processes

For any α∈(0,2), a truncated symmetric α-stable process in ℝ ^d is a symmetric Lévy process in ℝ ^d with no diffusion part and with a Lévy density given by c|x|^−d−α 1_{|x|<1} for some constant c. In (Kim and Song in Math. Z. 256(1): 139–173, [2007]) we have studied the potential theory of truncated symmetric stable processes. Among other things, we proved that the boundary Harnack principle is valid for the positive harmonic functions of this process in any bounded convex domain and showed that the Martin boundary of any bounded convex domain with respect to this process is the same as the Euclidean boundary. However, for truncated symmetric stable processes, the boundary Harnack principle is not valid in non-convex domains. In this paper, we show that, for a large class of not necessarily convex bounded open sets in ℝ ^d called bounded roughly connected κ-fat open sets (including bounded non-convex κ-fat domains), the Martin boundary with respect to any truncated symmetric stable process is still the same as the Euclidean boundary. We also show that, for truncated symmetric stable processes a relative Fatou type theorem is true in bounded roughly connected κ-fat open sets. The research of P. Kim is supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-331-C00037). The research of R. Song is supported in part by a joint US-Croatia grant INT 0302167.     

Polynomial-Time Algorithms for Multivariate Linear Problems with Finite-Order Weights: Average Case Setting

We study multivariate linear problems in the average case setting with respect to a zero-mean Gaussian measure whose covariance kernel has a finite-order weights structure. This means that the measure is concentrated on a Banach space of d-variate functions that are sums of functions of at most q ^* variables and the influence of each such term depends on a given weight. Here q ^* is fixed whereas d varies and can be arbitrarily large. For arbitrary finite-order weights, based on Smolyak’s algorithm, we construct polynomial-time algorithms that use standard information. That is, algorithms that solve the d-variate problem to within ε using of order function values modulo a power of ln ε ⁻¹. Here p is the exponent which measures the difficulty of the univariate (d=1) problem, and the power of ln ε ⁻¹ is independent of d. We also present a necessary and sufficient condition on finite-order weights for which we obtain strongly polynomial-time algorithms, i.e., when the number of function values is independent of d and polynomial in ε ⁻¹. The exponent of ε ⁻¹ may be, however, larger than p. We illustrate the results by two multivariate problems: integration and function approximation. For the univariate case we assume the r-folded Wiener measure. Then p=1/(r+1) for integration and for approximation.      

Optimization of convex functions on w*-compact sets

We give a direct, self-contained, and iterative proof that for any convex, Lipschitz andw ^*-lower semicontinuous function ϕ defined on aw ^*-compact convex setC in a dual Banach spaceX ^* and for any ε>0 there is anx∈X, with ‖x‖≤ε, such that ϕ+x attains its supremum at an extreme point ofC. This result is implicitly contained in the work of Lindenstrauss [9] and the work of Ghoussoub and Maurey on strongw ^*−H _σ sets [8]. In addition, we discuss the applications of this result to the geometry of convex sets. Research supported in part by the NSERC of Canada under grant OGP41983 for the first author and grant OGP7926 for the second author.     

PACKING CIRCLES OF TWO DIFFERENT SIZES ON THE SPHERE. II

The authors prove that the density of any packing of disks with radii r ₅ = π/5 and R ₅ = π/2 − π/5 cannot exceed the density of the incircles of the Archimedean tessellation of symbol (4,4,5). This revised version was published online in August 2006 with corrections to the Cover Date.     

Enumeration of three-dimensional convex polygons

A self-avoiding polygon (SAP) on a graph is an elementary cycle. Counting SAPs on the hypercubic lattice ℤ ^d withd≥2, is a well-known unsolved problem, which is studied both for its combinatorial and probabilistic interest and its connections with statistical mechanics. Of course, polygons on ℤ ^d are defined up to a translation, and the relevant statistic is their perimeter. A SAP on ℤ ^d is said to beconvex if its perimeter is “minimal”, that is, is exactly twice the sum of the side lengths of the smallest hyper-rectangle containing it. In 1984, Delest and Viennot enumerated convex SAPs on the square lattice [6], but no result was available in a higher dimension. We present an elementar approach to enumerate convex SAPs in any dimension. We first obtain a new proof of Delest and Viennot's result, which explains combinatorially the form of the generating function. We then compute the generating function for convex SAPs on the cubic lattice. In a dimension larger than 3, the details of the calculations become very cumbersome. However, our method suggests that the generating function for convex SAPs on ℤ ^d is always a quotient ofdifferentiably finite power series.     

On the negative Pell equation

Let ε be the fundamental unit of a field Q( ?d ) Q\left( {\sqrt {d} } \right) . In the paper, it is proved that ε > d ^3/2/ log² d for almost all d such that N(ε) = −1. Bibliography: 6 titles.     

Multiple points of trajectories of multiparameter fractional Brownian motion

Consider 0<α<1 and the Gaussian process Y(t) on ℝ ^N with covariance E(Y(s)Y(t))=|t|^2α+|s|^2α−|t−s|^2α, where |t| is the Euclidean norm of t. Consider independent copies X ¹,…,X ^d of Y and␣the process X(t)=(X ¹(t),…,X ^d (t)) valued in ℝ ^d . When kN≤␣(k−1)αd, we show that the trajectories of X do not have k-multiple points. If N<αd and kN>(k−1)αd, the set of k-multiple points of the trajectories X is a countable union of sets of finite Hausdorff measure associated with the function ϕ(ɛ)=ɛ ^{k N /α−( k −1) d} (loglog(1/ɛ)) ^k . If N=αd, we show that the set of k-multiple points of the trajectories of X is a countable union of sets of finite Hausdorff measure associated with the function ϕ(ɛ)=ɛ ^d (log(1/ɛ) logloglog 1/ɛ) ^k . (This includes the case k=1.) Received: 20 May 1997 / Revised version: 15 May 1998     

Contraction and Expansion of Convex Sets

Let S{\mathcal{S}} be a set system of convex sets in ℝ ^d . Helly’s theorem states that if all sets in S{\mathcal{S}} have empty intersection, then there is a subset S￠ ì S{\mathcal{S}}'\subset{\mathcal{S}} of size d+1 which also has empty intersection. The conclusion fails, of course, if the sets in S{\mathcal{S}} are not convex or if S{\mathcal{S}} does not have empty intersection. Nevertheless, in this work we present Helly-type theorems relevant to these cases with the aid of a new pair of operations, affine-invariant contraction, and expansion of convex sets. These operations generalize the simple scaling of centrally symmetric sets. The operations are continuous, i.e., for small ε>0, the contraction C ^−ε and the expansion C ^ε are close (in the Hausdorff distance) to C. We obtain two results. The first extends Helly’s theorem to the case of set systems with nonempty intersection: