Better lower bounds on detecting affine and spherical degeneracies

We show that in the worst case, Ω(n ^d ) sidedness queries are required to determine whether a set ofn points in ℝ ^d is affinely degenerate, i.e., whether it containsd+1 points on a common hyperplane. This matches known upper bounds. We give a straightforward adversary argument, based on the explicit construction of a point set containing Ω(n ^d ) “collapsible” simplices, any one of which can be made degenerate without changing the orientation of any other simplex. As an immediate corollary, we have an Ω(n ^d ) lower bound on the number of sidedness queries required to determine the order type of a set ofn points in ℝ ^d . Using similar techniques, we also show that Ω(n ^d+1) in-sphere queries are required to decide the existence of spherical degeneracies in a set ofn points in ℝ ^d . An earlier version of this paper was presented at the 34th Annual IEEE Symposium on Foundations of Computer Science [8]. This research has been supported by NSF Presidential Young Investigator Grant CCR-9058440.     

Strengthened semidefinite programming bounds for codes

We give a hierarchy of semidefinite upper bounds for the maximum size A(n,d) of a binary code of word length n and minimum distance at least d. At any fixed stage in the hierarchy, the bound can be computed (to an arbitrary precision) in time polynomial in n; this is based on a result of de Klerk et al. (Math Program, 2006) about the regular ∗-representation for matrix ∗-algebras. The Delsarte bound for A(n,d) is the first bound in the hierarchy, and the new bound of Schrijver (IEEE Trans. Inform. Theory 51:2859–2866, 2005) is located between the first and second bounds in the hierarchy. While computing the second bound involves a semidefinite program with O(n ⁷) variables and thus seems out of reach for interesting values of n, Schrijver’s bound can be computed via a semidefinite program of size O(n ³), a result which uses the explicit block-diagonalization of the Terwilliger algebra. We propose two strengthenings of Schrijver’s bound with the same computational complexity. Supported by the Netherlands Organisation for Scientific Research grant NWO 639.032.203.     

Line Transversals to Disjoint Balls

We prove that the set of directions of lines intersecting three disjoint balls in ℝ³ in a given order is a strictly convex subset of . We then generalize this result to n disjoint balls in ℝ ^d . As a consequence, we can improve upon several old and new results on line transversals to disjoint balls in arbitrary dimension, such as bounds on the number of connected components and Helly-type theorems.     

On visibility and covering by convex sets

A setX⊆ℝ ^d isn-convex if among anyn of its points there exist two such that the segment connecting them is contained inX. Perles and Shelah have shown that any closed (n+1)-convex set in the plane is the union of at mostn ⁶ convex sets. We improve their bound to 18n ³, and show a lower bound of order Ω(n ²). We also show that ifX⊆ℝ² is ann-convex set such that its complement has λ one-point path-connectivity components, λ<∞, thenX is the union ofO(n ⁴+n ²λ) convex sets. Two other results onn-convex sets are stated in the introduction (Corollary 1.2 and Proposition 1.4). Research supported by Charles University grants GAUK 99/158 and 99/159, and by U.S.-Czechoslovak Science and Technology Program Grant No. 94051. Part of the work by J. Matoušek was done during the author’s visits at Tel Aviv University and The Hebrew University of Jerusalem. Part of the work by P. Valtr was done during his visit at the University of Cambridge supported by EC Network DIMANET/PECO Contract No. ERBCIPDCT 94-0623.     

Bounding the number of connected components of a real algebraic set

For every polynomial mapf=(f ₁,…,f _k): ℝ ⁿ →ℝ ^k , we consider the number of connected components of its zero set,B(Z _f) and two natural “measures of the complexity off,” that is the triple(n, k, d), d being equal to max(degree off _i), and thek-tuple (Δ₁,...,Δ₄), Δ _k being the Newton polyhedron off _i respectively. Our aim is to boundB(Z _f) by recursive functions of these measures of complexity. In particular, with respect to (n, k, d) we shall improve the well-known Milnor-Thom’s bound μ _d (n)=d(2d−1) ⁿ⁻¹. Considered as a polynomial ind, μ _d (n) has leading coefficient equal to 2 ⁿ⁻¹. We obtain a bound depending onn, d, andk such that ifn is sufficiently larger thank, then it improves μ _d (n) for everyd. In particular, it is asymptotically equal to 1/2(k+1)n ^k−1 dⁿ, ifk is fixed andn tends to infinity. The two bounds are obtained by a similar technique involving a slight modification of Milnor-Thom's argument, Smith's theory, and information about the sum of Betti numbers of complex complete intersections.     

Multiwavelet packets and frame packets ofL 2(ℝ d )

The orthonormal basis generated by a wavelet ofL ²(ℝ) has poor frequency localization. To overcome this disadvantage Coifman, Meyer, and Wickerhauser constructed wavelet packets. We extend this concept to the higher dimensions where we consider arbitrary dilation matrices. The resulting basis ofL ²(ℝ ^d ) is called the multiwavelet packet basis. The concept of wavelet frame packet is also generalized to this setting. Further, we show how to construct various orthonormal bases ofL ²(ℝ ^d ) from the multiwavelet packets.     

Upper bounds for the diameter and height of graphs of convex polyhedra

Let Δ(d, n) be the maximum diameter of the graph of ad-dimensional polyhedronP withn-facets. It was conjectured by Hirsch in 1957 that Δ(d, n) depends linearly onn andd. However, all known upper bounds for Δ(d, n) were exponential ind. We prove a quasi-polynomial bound Δ(d, n)≤n ^{2 logd+3}. LetP be ad-dimensional polyhedron withn facets, let ϕ be a linear objective function which is bounded onP and letv be a vertex ofP. We prove that in the graph ofP there exists a monotone path leading fromv to a vertex with maximal ϕ-value whose length is at most . This research was supported in part by a BSF grant, by a GIF grant, and by ONR-N00014-91-C0026.     

Improved bounds on weak ε-nets for convex sets

LetS be a set ofn points in ℝ ^d . A setW is aweak ε-net for (convex ranges of)S if, for anyT⊆S containing εn points, the convex hull ofT intersectsW. We show the existence of weak ε-nets of size , whereβ ₂=0,β ₃=1, andβ _d ≈0.149·2 ^d-1(d-1)!, improving a previous bound of Alonet al. Such a net can be computed effectively. We also consider two special cases: whenS is a planar point set in convex position, we prove the existence of a net of sizeO((1/ε) log^1.6(1/ε)). In the case whereS consists of the vertices of a regular polygon, we use an argument from hyperbolic geometry to exhibit an optimal net of sizeO(1/ε), which improves a previous bound of Capoyleas. Work by Bernard Chazelle has been supported by NSF Grant CCR-90-02352 and the Geometry Center. Work by Herbert Edelsbrunner has been supported by NSF Grant CCR-89-21421. Work by Michelangelo Grigni has been supported by NSERC Operating Grants and NSF Grant DMS-9206251. Work by Leonidas Guibas and Micha Sharir has been supported by a grant from the U.S.-Israeli Binational Science Foundation. Work by Emo Welzl and Micha Sharir has been supported by a grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development. Work by Micha Sharir has also been supported by NSF Grant CCR-91-22103, and by a grant from the Fund for Basic Research administered by the Israeli Academy of Sciences.     

Steiner symmetrals and their distance from a ball

It is known that given any convex bodyK ⊂ ℝ ⁿ there is a sequence of suitable iterated Steiner symmetrizations ofK that converges, in the Hausdorff metric, to a ball of the same volume. Hadwiger and, more recently, Bourgain, Lindenstrauss and Milman have given estimates from above of the numberN of symmetrizations necessary to transformK into a body whose distance from the equivalent ball is less than an arbitrary positive constant. In this paper we will exhibit some examples of convex bodies which are “hard to make spherical”. For instance, for any choice of positive integersn≥2 andm, we construct ann-dimensional convex body with the property that any sequence ofm symmetrizations does not decrease its distance from the ball. A consequence of these constructions are some lower bounds on the numberN.     

Wavelet sets in ℝ n

A congruency theorem is proven for an ordered pair of groups of homeomorphisms of a metric space satisfying an abstract dilation-translation relationship. A corollary is the existence of wavelet sets, and hence of single-function wavelets, for arbitrary expansive matrix dilations on L ² (ℝ ⁿ). Moreover, for any expansive matrix dilation, it is proven that there are sufficiently many wavelet sets to generate the Borel structure ofℝ ⁿ. The second author is supported in part by NSF Grant DMS-9401544. The third author was a Graduate Research Assistant at Workshop in Linear Analysis and Probability, Texas A&M University.     

More Colourful Simplices

We show that any point in the convex hull of each of (d+1) sets of (d+1) points in general position in ℝ ^d is contained in at least ⌈(d+1)²/2⌉ simplices with one vertex from each set. This improves the known lower bounds for all d≥4.     

Voronoi diagrams of random lines and flats

It is proved that, for any fixedd ≽ 3 and 0 ≤k ≤ d - 1, the expected combinatorial complexity of the Euclidean Voronoi diagram ofn random &-flats drawn independently from the uniform distribution onk-flats intersecting the unit ball in ℝ^d is Ξ(n ^d/(d-k)) asn → ∞. A by-product of the proof is a density transformation for integrating over sets ofd + 1k-flats in ℝ^d     

Almost tight upper bounds for lower envelopes in higher dimensions

We consider the problem of bounding the combinatorial complexity of the lower envelope ofn surfaces or surface patches ind-space (d≥3), all algebraic of constant degree, and bounded by algebraic surfaces of constant degree. We show that the complexity of the lower envelope ofn such surface patches isO(n ^d−1+∈), for any ∈>0; the constant of proportionality depends on ∈, ond, ons, the maximum number of intersections among anyd-tuple of the given surfaces, and on the shape and degree of the surface patches and of their boundaries. This is the first nontrivial general upper bound for this problem, and it almost establishes a long-standing conjecture that the complexity of the envelope isO(n ^d-2λ _q (n)) for some constantq depending on the shape and degree of the surfaces (where λ _q (n) is the maximum length of (n, q) Davenport-Schinzel sequences). We also present a randomized algorithm for computing the envelope in three dimensions, with expected running timeO(n ^2+∈), and give several applications of the new bounds. Work on this paper has been supported by NSF Grant CCR-91-22103, and by grants from the U.S.-Israeli Binational Science Foundation, the G.I.F., the German-Israeli Foundation for Scientific Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences.     

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.     

Thed-step conjecture for polyhedra of dimensiond<6

Two functions Δ and Δ _b , of interest in combinatorial geometry and the theory of linear programming, are defined and studied. Δ(d, n) is the maximum diameter of convex polyhedra of dimensiond withn faces of dimensiond−1; similarly, Δ _b (d,n) is the maximum diameter of bounded polyhedra of dimensiond withn faces of dimensiond−1. The diameter of a polyhedronP is the smallest integerl such that any two vertices ofP can be joined by a path ofl or fewer edges ofP. It is shown that the boundedd-step conjecture, i.e. Δ _b (d,2d)=d, is true ford≤5. It is also shown that the generald-step conjecture, i.e. Δ(d, 2d)≤d, of significance in linear programming, is false ford≥4. A number of other specific values and bounds for Δ and Δ _b are presented. This revised version was published online in November 2006 with corrections to the Cover Date.     

On the Bethe-Sommerfeld conjecture for higher-order elliptic operators

 We consider the elliptic operator P(D)+V in ℝ ^d , d≥2 where P(D) is a constant coefficient elliptic pseudo-differential operator of order 2ℓ with a homogeneous convex symbol P(ξ), and V is a real periodic function in L ^∞(ℝ ^d ). We show that the number of gaps in the spectrum of P(D)+V is finite if 4ℓ>d+1. If in addition, V is smooth and the convex hypersurface {ξℝ ^d :P(ξ)=1} has positive Gaussian curvature everywhere, then the number of gaps in the spectrum of P(D)+V is finite, provided 8ℓ>d+3 and 9≥d≥2, or 4ℓ>d−3 and d≥10. Received: 10 October 2001 / Published online: 28 March 2003 Mathematics Subject Classification (1991): 35J10 Research supported in part by NSF Grant DMS-9732894.     

The expected value of some functions of the convex hull of a random set of points sampled inR d

This paper presents formulas and asymptotic expansions for the expected number of vertices and the expected volume of the convex hull of a sample ofn points taken from the uniform distribution on ad-dimensional ball. It is shown that the expected number of vertices is asymptotically proportional ton ^{(d−1)/(d+1)}, which generalizes Rényi and Sulanke’s asymptotic raten ^(1/3) ford=2 and agrees with Raynaud’s asymptotic raten ^{(d−1)/(d+1)} for the expected number of facets, as it should be, by Bárány’s result that the expected number ofs-dimensional faces has order of magnitude independent ofs. Our formulas agree with the ones Efron obtained ford=2 and 3 under more general distributions. An application is given to the estimation of the probability content of an unknown convex subset ofR ^d .     

On the number of minimal 1-Steiner trees

We count the number of nonisomorphic geometric minimum spanning trees formed by adding a single point to ann-point set ind-dimensional space, by relating it to a family of convex decompositions of space. TheO(n ^d log^{2d 2}−d n) bound that we obtain significantly improves previously known bounds and is tight to within a polylogarithmic factor. The research of D. Eppstein was performed in part while visiting Xerox PARC.     

Proof of the double bubble curvature conjecture

An area minimizing double bubble in ℝⁿ is given by two (not necessarily connected) regions which have two prescribed n-dimensional volumes whose combined boundary has least (n−1)-dimensional area. The double bubble theorem states that such an area minimizer is necessarily given by a standard double bubble, composed of three spherical caps. This has now been proven for n = 2, 3,4, but is, for general volumes, unknown for n ≥ 5. Here, for arbitrary n, we prove a conjectured lower bound on the mean curvature of a standard double bubble. This provides an alternative line of reasoning for part of the proof of the double bubble theorem in ℝ³, as well as some new component bounds in ℝⁿ.     

Low dimensional sections versus projections of convex bodies

The structure of low dimensional sections and projections of symmetric convex bodies is studied. For a symmetric convex bodyB ⊂ ℝ ⁿ , inequalities between the smallest diameter of rank ℓ projections ofB and the largest in-radius ofm-dimensional sections ofB are established, for a wide range of sub-proportional dimensions. As an application it is shown that every bodyB in (isomorphic) ℓ-position admits a well-bounded (√n, 1)-mixing operator. Research of this author was partially supported by KBN Grant no. 1 P03A 015 27. This author holds the Canada Research Chair in Geometric Analysis.