On neighborly families of convex bodies

A family of convex bodies in E^d is called neighborly if the intersection of every two of them is (d-1)-dimensional. In the present paper we prove that there is an infinite neighborly family of centrally symmetric convex bodies in E^d, d 3, such that every two of them are affinely equivalent (i.e., there is an affine transformation mapping one of them onto another), the bodies have large groups of affine automorphisms, and the volumes of the bodies are prescribed. We also prove that there is an infinite neighborly family of centrally symmetric convex bodies in E^d such that the bodies have large groups of symmetries. These two results are answers to a problem of B. Grünbaum (1963). We prove also that there exist arbitrarily large neighborly families of similar convex d-polytopes in E^d with prescribed diameters and with arbitrarily large groups of symmetries of the polytopes.     

Projection problems for symmetric polytopes

Let K⊂Rn be a convex body (a compact, convex subset with non-empty interior), ΠK its projection body. Finding the least upper bound, as K ranges over the class of origin-symmetric convex bodies, of the affine-invariant ratio V(ΠK)/V(K)ⁿ⁻¹, being called Schneider's projection problem, is a well-known open problem in the convex geometry. To study this problem, Lutwak, Yang and Zhang recently introduced a new affine invariant functional for convex polytopes in Rn. For origin-symmetric convex polytopes, they posed a conjecture for the new functional U(P). In this paper, we give an affirmative answer to the conjecture in Rn, thereby, obtain a modified version of Schneider's projection problem.     

Constant Minkowskian width in terms of double normals

We extend the notion of a double normal of a convex body from smooth, strictly convex Minkowski planes to arbitrary two-dimensional real, normed, linear spaces in two different ways. Then, for both of these ways, we obtain the following characterization theorem: a convex body K in a Minkowski plane is of constant Minkowskian width iff every chord I of K splits K into two compact convex sets K₁ and K₂ such that I is a Minkowskian double normal of K₁ or K₂. Furthermore, the Euclidean version of this theorem yields a new characterization of d-dimensional Euclidean ball where d 3.     

Mean projections and finite packings of convex bodies

Consider the convex hullQ ofn non-overlapping translates of a convex bodyC inE ^d ,n be large. IfQ has minimali-dimensional projection, 1i<d then we prove thatQ is approximately a sphere.     

The cross-section body, plane sections of convex bodies and approximation of convex bodies, I

For a convex body K ^d we investigate three associated bodies, its intersection body IK (for 0int K), cross-section body CK, and projection body IIK, which satisfy IKCKIIK. Conversely we prove CKconst₁(d)I(K–x) for some xint K, and IIKconst₂ (d)CK, for certain constants, the first constant being sharp. We estimate the maximal k-volume of sections of 1/2(K+(-K)) with k-planes parallel to a fixed k-plane by the analogous quantity for K; our inequality is, if only k is fixed, sharp. For L ^d a convex body, we take n random segments in L, and consider their Minkowski average D. We prove that, for V(L) fixed, the supremum of V(D) (with also nN arbitrary) is minimal for L an ellipsoid. This result implies the Petty projection inequality about max V((IIM)*), for M ^d a convex body, with V(M) fixed. We compare the volumes of projections of convex bodies and the volumes of the projections of their sections, and, dually, the volumes of sections of convex bodies and the volumes of sections of their circumscribed cylinders. For fixed n, the pth moments of V(D) (1p<) also are minimized, for V(L) fixed, by the ellipsoids. For k=2, the supremum (nN arbitrary) and the pth moment (n fixed) of V(D) are maximized for example by triangles, and, for L centrally symmetric, for example by parallelograms. Last we discuss some examples for cross-section bodies.Research (partially) supported by Hungarian National Foundation for Scientific Research, Grant No. 41.     

Variations on the theme of repeated distances

We give an asymptotically sharp estimate for the error term of the maximum number of unit distances determined byn points in ^{d, d}4. We also give asymptotically tight upper bounds on the total number of occurrences of the favourite distances fromn points in ^{d, d}4. Related results are proved for distances determined byn disjoint compact convex sets in ².At the time this paper was written, both authors were visiting the Technion — Israel Institute of Technology.     

On a question of K. Bezdek and T. Odor on Γ-parallelotops

K. Bezdek and T. Odor proved the following statement in [1]: If a covering ofE ³ is a lattice packing of the convex compact bodyK with packing lattice Λ (K is a Λ-parallelotopes) then there exists such a 2-dimensional sublattice Λ′ of Λ which is covered by the set ∪(K+z∣z ∈ Λ′). (K ∪L(Λ′) is a Λ′-parallelotopes). We prove that the statement is not true in the case of the dimensionsn=6, 7, 8. Supported by Hung. Nat. Found for Sci. Research (OTKA) grant no. 1615 (1991).     

Expectation of random polytopes

A random polytopeP _n in a convex bodyC is the convex hull ofn identically and independently distributed points inC. Its expectation is a convex body in the interior ofC. We study the deviation of the expectation ofP _n fromC asn→∞: while forC of classC ^k+1,k≥1, precise asymptotic expansions for the deviation exist, the behaviour of the deviation is extremely irregular for most convex bodiesC of classC ¹. Dedicated to my teacher and friend Professor Edmund Hlawka on the occasion of his 80^th birthday     

Highly saturated packings and reduced coverings

We introduce and study certain notions which might serve as substitutes for maximum density packings and minimum density coverings. A body is a compact connected set which is the closure of its interior. A packingP with congruent replicas of a bodyK isn-saturated if non–1 members of it can be replaced withn replicas ofK, and it is completely saturated if it isn-saturated for eachn1. Similarly, a coveringC with congruent replicas of a bodyK isn-reduced if non members of it can be replaced byn–1 replicas ofK without uncovering a portion of the space, and its is completely reduced if it isn-reduced for eachn1. We prove that every bodyK ind-dimensional Euclidean or hyperbolic space admits both ann-saturated packing and ann-reduced covering with replicas ofK. Under some assumptions onKE ^d (somewhat weaker than convexity), we prove the existence of completely saturated packings and completely reduced coverings, but in general, the problem of existence of completely saturated packings, and completely reduced coverings remains unsolved. Also, we investigate some problems related to the the densities ofn-saturated packings andn-reduced coverings. Among other things, we prove that there exists an upper bound for the density of ad+2-reduced covering ofE ^d with congruent balls, and we produce some density bounds for then-saturated packings andn-reduced coverings of the plane with congruent circles.     

Common supports as fixed points

A family S of sets in R ^d is sundered if for each way of choosing a point from rd+1 members of S, the chosen points form the vertex-set of an (r–1)-simplex. Bisztriczky proved that for each sundered family S of d convex bodies in R ^d , and for each partition (S , S ), of S, there are exactly two hyperplanes each of which supports all the members of S and separates the members of S from the members of S . This note provides an alternate proof by obtaining each of the desired supports as (in effect) a fixed point of a continuous self-mapping of the cartesian product of the bodies.     

Volume approximation of smooth convex bodies by three-polytopes of restricted number of edges

For a given convex body K in with C ² boundary, let P ^c _n be the circumscribed polytope of minimal volume with at most n edges, and let P ⁱ _n be the inscribed polytope of maximal volume with at most n edges. Besides presenting an asymptotic formula for the volume difference as n tends to infinity in both cases, we prove that the typical faces of P ^c _n and P ⁱ _n are asymptotically regular triangles and squares, respectively, in a suitable sense. Supported by OTKA grants 043520 and 049301, and by the EU Marie Curie grants Discconvgeo, Budalggeo and PHD. Authors’ addresses: Károly J. B?r?czky, Alfréd Rényi Institute of Mathematics, P.O. Box 127, Budapest H–1364, Hungary, and Department of Geometry, Roland E?tv?s University, Pázmány Péter sétány 1/C, Budapest 1117, Hungary; Salvador S. Gomis, Department of Mathematical Analysis, University of Alicante, 03080 Alicante, Spain; Péter Tick, Gyűrű utca 24, Budapest H–1039, Hungary     

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.     

On the Sobolev distance of convex bodies

Summary LetK ^d denote the cone of all convex bodies in the Euclidean spaceK ^d . The mappingK h _K of each bodyK K ^d onto its support function induces a metric _w onK ^d by" _w (K, L)h _L –h _{K w} where _w is the Sobolev I-norm on the unit sphere . We call _w (K, L) the Sobolev distance ofK andL. The goal of our paper is to develop some fundamental properties of the Sobolev distance.     

On the volume of convex hulls of sets on spheres

We prove that for a measurable subset of S ^n–1 with fixed Haar measure, the volume of its convex hull is minimized for a cap (i.e. a ball with respect to the geodesic measure). We solve a similar problem for symmetric sets and n=2, 3. As a consequence, we deduce a result concerning Gaussian measures of dilatations of convex, symmetric sets in R ² and R ³.Partially supported by KBN (Poland), Grant No. 2 1094 91 01.     

Cohomology vanishing¶and a problem in approximation theory

For a simplicial subdivison Δ of a region in k ⁿ (k algebraically closed) and r∈N, there is a reflexive sheaf ? on P ⁿ , such that H ⁰(?(d)) is essentially the space of piecewise polynomial functions on Δ, of degree at most d, which meet with order of smoothness r along common faces. In [9], Elencwajg and Forster give bounds for the vanishing of the higher cohomology of a bundle ℰ on P ⁿ in terms of the top two Chern classes and the generic splitting type of ℰ. We use a spectral sequence argument similar to that of [16] to characterize those Δ for which ? is actually a bundle (which is always the case for n= 2). In this situation we can obtain a formula for H ⁰(?(d)) which involves only local data; the results of [9] cited earlier allow us to give a bound on the d where the formula applies. We also show that a major open problem in approximation theory may be formulated in terms of a cohomology vanishing on P ² and we discuss a possible connection between semi-stability and the conjectured answer to this open problem. Received: 9 April 2001     

On the smallest disk containing a planar reduced convex body

A convex body R of Euclidean space E ^d is said to be reduced if every convex body $ P \subset R $ different from R has thickness smaller than the thickness $ \Delta(R) $ of R. We prove that every planar reduced body R is contained in a disk of radius $ {1\over 2}\sqrt 2 \cdot \Delta(R) $. For $ d \geq 3 $, an analogous property is not true because we can construct reduced bodies of thickness 1 and of arbitrarily large diameter.     

Characterization off-vectors of families of convex sets inR d Part I: Necessity of Eckhoff’s conditions

LetK=K ₁,...,K_n be a family ofn convex sets inR ^d . For 0≦i<n denote byf _i the number of subfamilies ofK of sizei+1 with non-empty intersection. The vectorf(K) is called thef-vectors ofK. In 1973 Eckhoff proposed a characterization of the set off-vectors of finite families of convex sets inR ^d by a system of inequalities. Here we prove the necessity of Eckhoff's inequalities. The proof uses exterior algebra techniques. We introduce a notion of generalized homology groups for simplicial complexes. These groups play a crucial role in the proof, and may be of some independent interest.     

Geometric methods in the study of irregularities of distribution

Let be a signed measure on E ^d with E ^d =0 and ¦¦E^d<. DefineD _s() as sup ¦H¦ whereH is an open halfspace. Using integral and metric geometric techniques results are proved which imply theorems such as the following.Theorem A. Let be supported by a finite pointsetp _i. ThenD _s()>c _d(₁/ ₂)^1/2{ _{i(p i})²}^1/2 where ₁ is the minimum distance between two distinctp _i, and ₂ is the maximum distance. The numberc _d is an absolute dimensional constant. (The number .05 can be chosen forc ₂ in Theorem A.)Theorem B. LetD be a disk of unit area in the planeE ², andp ₁,p ₂,...,p _n be a set of points lying inD. If m if the usual area measure restricted toD, while nP _i=1/n defines an atomic measure _n, then independently of _n,nD _s(m – _n) .0335n ^1/4. Theorem B gives an improved solution to the Roth disk segment problem as described by Beck and Chen. Recent work by Beck shows thatnD _s(m – _n)cn ^1/4(logn)^–7/2.     

Directeds-t numberings,Rubber bands,and testing digraphk-vertex connectivity

LetG=(V, E) be a directed graph andn denote |V|. We show thatG isk-vertex connected iff for every subsetX ofV with |X| =k, there is an embedding ofG in the (k–1)-dimensional spaceR ^k–1,fVR ^k–1, such that no hyperplane containsk points of {f(v)|vV}, and for eachvV–X, f(v) is in the convex hull of {f(w)| (v, w)E}. This result generalizes to directed graphs the notion of convex embeddings of undirected graphs introduced by Linial, Lovász and Wigderson in Rubber bands, convex embeddings and graph connectivity,Combinatorica 8 (1988), 91–102.Using this characterization, a directed graph can be tested fork-vertex connectivity by a Monte Carlo algorithm in timeO((M(n)+nM(k)) · (logn)) with error probability<1/n, and by a Las Vegas algorithm in expected timeO((M(n)+nM(k)) ·k), whereM(n) denotes the number of arithmetic steps for multiplying twon×n matrices (M(n)=O(n ^2.376)). Our Monte Carlo algorithm improves on the best previous deterministic and randomized time complexities fork>n ^0.19; e.g., for , the factor of improvement is >n ^0.62. Both algorithms have processor efficient parallel versions that run inO((logn)²) time on the EREW PRAM model of computation, using a number of processors equal to logn times the respective sequential time complexities. Our Monte Carlo parallel algorithm improves on the number of processors used by the best previous (Monte Carlo) parallel algorithm by a factor of at leastn ²/(logn)³ while having the same running time.Generalizing the notion ofs-t numberings, we give a combinatorial construction of a directeds-t numbering for any 2-vertex connected directed graph.     

Subdivision algorithms and the kernel of a polyhedron

The kernelK of a convex polyhedronP ₀, as defined by L. Fejes Tóth, is the limit of the sequence (P _n), whereP _n is the convex hull of the midpoints of the edges ofP _n−1. The boundary ∂K of the convex bodyK is investigated. It is shown that ∂K contains no two-dimensional faces and that ∂K need not belong toC ². The connection with similar algorithms from CAD (computer aided design) is explained and utilized. R. J. Gardner was supported in part by a von Humboldt fellowship.