Generalized Ham-Sandwich Cuts

Bárány, Hubard, and Jerónimo recently showed that for given well-separated convex bodies S ₁,…,S _d in R ^d and constants β _i ∈[0,1], there exists a unique hyperplane h with the property that Vol (h ⁺∩S _i )=β _i ⋅Vol (S _i ); h ⁺ is the closed positive transversal halfspace of h, and h is a “generalized ham-sandwich cut.” We give a discrete analogue for a set S of n points in R ^d which are partitioned into a family S=P ₁∪⋅⋅⋅∪P _d of well-separated sets and are in weak general position. The combinatorial proof inspires an O(n(log n) ^d−3) algorithm which, given positive integers a _i ≤|P _i |, finds the unique hyperplane h incident with a point in each P _i and having |h ⁺∩P _i |=a _i . Finally we show two other consequences of the direct combinatorial proof: the first is a stronger result, namely that in the discrete case, the conditions assuring existence and uniqueness of generalized cuts are also necessary; the second is an alternative and simpler proof of the theorem in Bárány et al., and in addition, we strengthen the result via a partial converse.     

Expectation of intrinsic volumes of random polytopes

Let K be a convex body in ℝ ^d , let j ∈ {1, …, d−1}, and let K(n) be the convex hull of n points chosen randomly, independently and uniformly from K. If ∂K is C ₊², then an asymptotic formula is known due to M. Reitzner (and due to I. Bárány if ∂K is C ₊³) for the difference of the jth intrinsic volume of K and the expectation of the jth intrinsic volume of K(n). We extend this formula to the case when the only condition on K is that a ball rolls freely inside K. Funded by the Marie-Curie Research Training Network “Phenomena in High-Dimensions” (MRTN-CT-2004-511953).     

The hypermetric cone is polyhedral

The hypermetric coneH _n is the cone in the spaceR ^n(n–1)/2 of all vectorsd=(d _ij)_1i<jn satisfying the hypermetric inequalities: –_1ijn z _j z _j d _ij 0 for all integer vectorsz inZ ⁿ with –_1in z _i =1. We explore connections of the hypermetric cone with quadratic forms and the geometry of numbers (empty spheres andL-polytopes in lattices). As an application, we show that the hypermetric coneH _n is polyhedral.     

Large antipodal families

A family {A _i | i ∈ I} of sets in ℝ ^d is antipodal if for any distinct i, j ∈ I and any p ∈ A _i , q ∈ A _j , there is a linear functional ϕ:ℝ ^d → ℝ such that ϕ(p) ≠ ϕ(q) and ϕ(p) ≤ ϕ(r) ≤ ϕ(q) for all r ∈ ∪ _i∈I A _i . We study the existence of antipodal families of large finite or infinite sets in ℝ³. The research was supported by the Hungarian-South African Intergovernmental Scientific and Technological Cooperation Programme, NKTH Grant no. ZA-21/2006 and South African National Research Foundation Grant no. UID 61853, as well as Hungarian National Foundation for Scientific Research Grants no. NK 67867, no. T47102, and no. K72537.     

Dependent percolation in two dimensions

For a natural number k, define an oriented site percolation on ℤ² as follows. Let x _i , y _j be independent random variables with values uniformly distributed in {1, …, k}. Declare a site (i, j) ∈ℤ² closed if x _i = y _j , and open otherwise. Peter Winkler conjectured some years ago that if k≥ 4 then with positive probability there is an infinite oriented path starting at the origin, all of whose sites are open. I.e., there is an infinite path P = (i ₀, j ₀)(i ₁, j ₁) · · · such that 0 = i ₀≤i ₁≤· · ·, 0 = j ₀≤j ₁≤· · ·, and each site (i _n , j _n ) is open. Rather surprisingly, this conjecture is still open: in fact, it is not known whether the conjecture holds for any value of k. In this note, we shall prove the weaker result that the corresponding assertion holds in the unoriented case: if k≤ 4 then the probability that there is an infinite path that starts at the origin and consists only of open sites is positive. Furthermore, we shall show that our method can be applied to a wide variety of distributions of (x _i ) and (y _j ). Independently, Peter Winkler [14] has recently proved a variety of similar assertions by different methods. Received: 4 March 1999 / Revised version: 27 September 1999 / Published online: 21 June 2000     

Reconstructing dimensions of intersections of convex sets

Denoting by dimA the dimension of the affine hull of the setA, we prove that if {K _i:i T} and {K _i ^j :i T} are two finite families of convex sets inR ⁿ and if dim {K _i :i S} = dim {K _i ^j :i S}for eachS T such that|S| n + 1 then dim {K _i :i T} = dim {K _i : {i T}}.     

Krasnosel'skii numbers for bounded finitely starlike sets inR d

For eachk andd, 1kd, definef(d, d)=d+1 andf(d, k)=2d if 1kd–1. The following results are established:Let be a uniformly bounded collection of compact, convex sets inR ^d . For a fixedk, 1kd, dim {MM in }k if and only if for some > 0, everyf(d, k) members of contain a commonk-dimensional set of measure (volume) at least.LetS be a bounded subset ofR ^d . Assume that for some fixedk, 1kd, there exists a countable family of (k–l)-flats {H _i :i1} inR ^d such that clS S {H_i i 1 } and for eachi1, (clS S) H _i has (k–1) dimensional measure zero. Every finite subset ofS sees viaS a set of positivek-dimensional measure if and only if for some>0, everyf(d,k) points ofS see viaS a set ofk-dimensional measure at least .The numbers off(d,d) andf(d, 1) above are best possible.Supported in part by NSF grant DMS-8705336.     

A remark on the unitary group of a tensor product of<Emphasis Type="Italic">n</Emphasis> finite-dimensional Hilbert spaces

LetH _i, 1 ≤ i ≤n be complex finite-dimensional Hilbert spaces of dimension d_i,1 ≤ i ≤n respectively withd _i ≥ 2 for everyi. By using the method of quantum circuits in the theory of quantum computing as outlined in Nielsen and Chuang [2] and using a key lemma of Jaikumar [1] we show that every unitary operator on the tensor productH =H ₁ ⊗H ₂ ⊗... ⊗H _n can be expressed as a composition of a finite number of unitary operators living on pair productsH _i ⊗H _j,1 ≤i,j ≤n. An estimate of the number of operators appearing in such a composition is obtained. Dedicated to Prof. A.K. Roy on his 62nd birthday.     

Directed Projection Functions of Convex Bodies

For 1 ≤ i < j < d, a j-dimensional subspace L of and a convex body K in , we consider the projection K|L of K onto L. The directed projection function v _i,j (K;L,u) is defined to be the i-dimensional size of the part of K|L which is illuminated in direction u ∈ L. This involves the i-th surface area measure of K|L and is motivated by Groemer’s [17] notion of semi-girth of bodies in . It is well-known that centrally symmetric bodies are determined (up to translation) by their projection functions, we extend this by showing that an arbitrary body is determined by any one of its directed projection functions. We also obtain a corresponding stability result. Groemer [17] addressed the case i = 1, j = 2, d = 3. For j > 1, we then consider the average of v _1,j (K;L,u) over all spaces L containing u and investigate whether the resulting function determines K. We will find pairs (d,j) for which this is the case and some pairs for which it is false. The latter situation will be seen to be related to some classical results from number theory. We will also consider more general averages for the case of centrally symmetric bodies. The research of the first author was supported in part by NSF Grant DMS-9971202 and that of the second author by a grant from the Volkswagen Foundation.     

Packing nearly-disjoint sets

De Bruijn and Erdős proved that ifA ₁, ...,A _k are distinct subsets of a set of cardinalityn, and |A _i ∩A _j |≦1 for 1≦i<j ≦k, andk>n, then some two ofA ₁, ...,A _k have empty intersection. We prove a strengthening, that at leastk /n ofA ₁, ...,A _k are pairwise disjoint. This is motivated by a well-known conjecture of Erdőds, Faber and Lovász of which it is a corollary. Partially supported by N. S. F. grant No. MCS—8103440     

Characterizing compact unions of two starshaped sets inR d

SetS inR ^d has propertyK ₂ if and only ifS is a finite union ofd-polytopes and for every finite setF in bdryS there exist points c₁,c₂ (depending onF) such that each point ofF is clearly visible viaS from at least one c_i,i = 1,2. The following characterization theorem is established: Let , d2. SetS is a compact union of two starshaped sets if and only if there is a sequence {S _j } converging toS (relative to the Hausdorff metric) such that each setS _j satisfies propertyK ₂. For , the sufficiency of the condition above still holds, although the necessity fails.     

Visible shorelines inR d

Summary LetC be a closed set inR ^d and letj be a fixed integer,j 1. The setS R ^d ~C is said to have aj-partition relative toC if there existj or fewer pointsc ₁,, c _j ofC such that each point ofS sees via the complement ofC at least one pointc _i. For every triple of integersd, p, j withd 0, p d + 1, j 1, there exists a smallest integerf(d, p, j) such that the following is true: IfC is a convexd-polytope inR ^d havingp vertices and ifS R ^d ~C, S has aj-partition relative toC if and only if everyf(d, p, j)-member subset of S has such a partition.ForC a convex polytope inR ² andS R ² ~C, all points ofS see via the complement ofC a common neighborhood in the boundary ofC if and only if every three points ofS see via the complement ofC such a neighborhood.A weak analogue of this result holds for arbitrary compact convex sets inR ^d .     

Carathéodory-type Theorems à la Bárány

We generalise the famous Helly–Lovász theorem leading to a generalisation of the Bárány–Carathéodory theorem for oriented matroids in dimension ≤3. We also provide a non-metric proof of the latter colourful theorem for arbitrary dimensions and explore some generalisations in dimension 2.     

Systems of equations driven by stable processes

Let Z^j_t, j = 1, . . . , d, be independent one-dimensional symmetric stable processes of index α ∈ (0,2). We consider the system of stochastic differential equations where the matrix A(x)=(A_ij(x))1≤ i, j ≤ d is continuous and bounded in x and nondegenerate for each x. We prove existence and uniqueness of a weak solution to this system. The approach of this paper uses the martingale problem method. For this, we establish some estimates for pseudodifferential operators with singular state-dependent symbols. Let λ₂ > λ₁ > 0. We show that for any two vectors a, b∈ ℝ_d with |a|, |b| ∈ (λ₁, λ₂) and p sufficiently large, the L^p-norm of the operator whose Fourier multiplier is (|u · a|^α - |u · b|^α) / ∑_j=1^d |u_i|^α is bounded by a constant multiple of |a−b|^θ for some θ > 0, where u=(u₁ , . . . , u_d) ∈ ℝ^d. We deduce from this the L^p-boundedness of pseudodifferential operators with symbols of the form ψ(x,u)=|u · a(x)|^α / ∑_j=1^d |u_i|^α, where u=(u₁,...,u_d) and a is a continuous function on ℝ^d with |a(x)|∈ (λ₁, λ₂) for all x∈ ℝ^d. Research partially supported by NSF grant DMS-0244737. Research partially supported by NSF grant DMS-0303310.     

The homology and cohomology of the complements to some arrangements of codimension two complex planes

We consider the homology and cohomology of the complement to the arrangement Z = ∪_{1<|i−j|<d−1}{z _i = z _j = 0} of coordinate planes in ℂ ^d , and explicitly construct a basis for these groups as well as a basis for the homology groups of the one-point compactification of Z.     

Analogues of Horn’s theorem for finite unions of starshaped sets in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

Fix k, d, 1 ≤ k ≤ d + 1. Let $ \mathcal{F} $ \mathcal{F} be a nonempty, finite family of closed sets in ℝ ^d , and let L be a (d − k + 1)-dimensional flat in ℝ ^d . The following results hold for the set T ≡ ∪{F: F in $ \mathcal{F} $ \mathcal{F} }. Assume that, for every k (not necessarily distinct) members F ₁, …, F _k of $ \mathcal{F} $ \mathcal{F} ,∪{F _i : 1 ≤ i ≤ k} is starshaped and the corresponding kernel contains a translate of L. Then T is starshaped, and its kernel also contains a translate of L.     

A Geometric Proof of the Colored Tverberg Theorem

The colored Tverberg theorem asserts that for every d and r there exists t=t(d,r) such that for every set C⊂ℝ ^d of cardinality (d+1)t, partitioned into t-point subsets C ₁,C ₂,…,C _d+1 (which we think of as color classes; e.g., the points of C ₁ are red, the points of C ₂ blue, etc.), there exist r disjoint sets R ₁,R ₂,…,R _r ⊆C that are rainbow, meaning that |R _i ∩C _j |≤1 for every i,j, and whose convex hulls all have a common point.     

Canonical Theorems for Convex Sets

Let F be a family of pairwise disjoint compact convex sets in the plane such that none of them is contained in the convex hull of two others, and let r be a positive integer. We show that F has r disjoint ⌊ c _r n⌋ -membered subfamilies F _i (1 ≤ i ≤ r) such that no matter how we pick one element F _i from each F _i , they are in convex position, i.e., every F _i appears on the boundary of the convex hull of ⋃ _i=1 ^r F _i . (Here c _r is a positive constant depending only on r .) This generalizes and sharpens some results of Erdős and Szekeres, Bisztriczky and Fejes Tóth, Bárány and Valtr, and others. <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p427.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader> Received April 30, 1997, and in revised form August 5, 1997.     

Stabbing Simplices by Points and Flats

The following result was proved by Bárány in 1982: For every d≥1, there exists c _d >0 such that for every n-point set S in ℝ ^d , there is a point p∈ℝ ^d contained in at least c _d n ^d+1−O(n ^d ) of the d-dimensional simplices spanned by S. We investigate the largest possible value of c _d . It was known that c _d ≤1/(2 ^d (d+1)!) (this estimate actually holds for every point set S). We construct sets showing that c _d ≤(d+1)^−(d+1), and we conjecture that this estimate is tight. The best known lower bound, due to Wagner, is c _d ≥γ _d :=(d ²+1)/((d+1)!(d+1) ^d+1); in his method, p can be chosen as any centerpoint of S. We construct n-point sets with a centerpoint that is contained in no more than γ _d n ^d+1+O(n ^d ) simplices spanned by S, thus showing that the approach using an arbitrary centerpoint cannot be further improved.     

The size of the shadow boundary projection

Among all embedded closed manifoldsM ^d ⊂ ℝ ^d+1 with positive exterior curvature ≤k the ratio between the (d − 1)-Hausdorff measure of the shadow boundary projection and the volume ofM ^d is maximized by the sphere of radius 1/k.