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.     

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.     

Improved Krasnosel'skii theorems for the dimension of the kernel of a starshaped set

We will establish the following improved Krasnosel'skii theorems for the dimension of the kernel of a starshaped set: For each k and d, 0 k d, define f(d,k) = d+1 if k = 0 and f(d,k) = max{d+1,2d–2k+2} if 1 k d.Theorem 1. Let S be a compact, connected, locally starshaped set in R^d, S not convex. Then for a k with 0 k d, dim ker S k if and only if every f(d, k) lnc points of S are clearly visible from a common k-dimensional subset of S.Theorem 2. Let S be a nonempty compact set in R^d. Then for a k with 0 k d, dim ker S k if and only if every f (d, k) boundary points of S are clearly visible from a common k-dimensional subset of S. In each case, the number f(d, k) is best possible for every d and k.     

An Analogue of Hajós' Theorem for the Circular Chromatic Number (II)

This paper designs a set of graph operations, and proves that for 2k/d<3, starting from K_k/_d, by repeatedly applying these operations, one can construct all graphs G with _c(G)k/d. Together with the result proved in [20], where a set of graph operations were designed to construct graphs G with _c(G)k/d for k/d3, we have a complete analogue of Hajós' Theorem for the circular chromatic number. This research was partially supported by the National Science Council under grant NSC 89-2115-M-110-003     

The steinitz lemma inl inf∞ sup2

Let (a, b) be a pair of non-negative numbers such that (1)a, b1 and (2)a+b3. Letu ₁,...,u _n be a sequence of vectors from the set {(x, y)R ²: |x|, |y|1}, withu ₁+...+u _n =0. It is shown that there is a permutation of indices such that all partial sumsu ₍₁₎+...+u _(k) lie in the rectangle |x|a, |y|b. Conditions (1) and (2) are also necessary.     

The caratheodory number for thek-core

Thek-core of the setS ⁿ is the intersection of the convex hull of all setsA S with ¦SA¦<-k. The Caratheodory number of thek-core is the smallest integerf (d,k) with the property thatx core _{kS, S} ⁿ implies the existence of a subsetT S such thatx core_kT and ¦T¦f (d, k). In this paper various properties off(d, k) are established.Research of this author was partially supported by Hungarian National Science Foundation grant no. 1812.     

A Helly-type theorem for countable intersections of starshaped sets

Let k and d be fixed integers, 0kd, and let be a collection of sets in If every countable subfamily of has a starshaped intersection, then is (nonempty and) starshaped as well. Moreover, if every countable subfamily of has as its intersection a starshaped set whose kernel is at least k-dimensional, then the kernel of is at least k-dimensional, too. Finally, dual statements hold for unions of sets.Received: 3 April 2004     

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 convergence subsystems of orthonormal systems

It is proved that for any sequence {R _k} _k=1 of real numbers satisfyingR _kk (k1) andR _k=o(k log₂ k),k, there exists an orthonormal system {n _k(x)} _n=1 ,x (0;1), such that none of its subsystems {n _k(x)} _k=1 withn _kR_k (k1) is a convergence subsystem.     

On the Achievement of the Griesmer Bound

The main theorem in this paper is that there does not exist an [n,k,d]_q code with d = (k-2)q ^k-1 - (k-1)q^k-2 attaining the Griesmer bound for q k, k=3,4,5 and for q 2k-3, k 6.     

Some Erdös-Szekeres type results about points in space

The Erdös-Szekeres convexn-gon theorem states that for anyn3, there is a smallest integerf(n) such that any set of at leastf(n) points in the planeE ², no three collinear, contains the vertices of a convexn-gon. We consider three versions of this result as applied to convexly independent points and convex polytopes inE ^d >,d2.     

Some Applications of Critical Point Theory of Distance Functions on Riemannian Manifolds

In this paper, we use the theory of critical points of distance functions to study the rigidity and topology of Riemannian manifolds with sectional curvature bounded below. We prove that an n-dimensional complete connected Riemannian manifold M with sectional curvature K _M 1 is isometric to an n-dimensional Euclidean unit sphere if M has conjugate radius bigger than /2 and contains a geodesic loop of length 2. We also prove that if M is an n(3)-dimensional complete connected Riemannian manifold with K _M 1 and radius bigger than /2, then any closed connected totally geodesic submanifold of dimension not less than two of M is homeomorphic to a sphere.     

Some tilings of the plane whose singular points are uncountable and unbounded

Let I be a tiling of the plane such that for every tile T of I there correspond a tile T of I (not necessarily unique) and an integer k(T, T) (depending on T and T), k(T, T)>2, such that T meets T in k(T, T) connected components. Tiles T and T satisfying this condition are called associated tiles in I. Various properties concerning I and its singular points are obtained. First, it is not possible that every tile in I have a unique associated tile. In fact, there exist infinite families of tiles {F} {F _n:n1} such that F is the unique associated tile for every F _n. Next, if x is a singular point of I, then every neighborhood of x contains uncountably many singular points of I. Finally, the set of singular points of I is unbounded.     

Light paths with an odd number of vertices in large polyhedral maps

LetP _k be a path onk vertices. In this paper we prove that (1) every polyhedral map on the torus and the Klein bottle contains a pathP _k such that each of its vertices has degree 6k–2 ifk is odd,k3, (2) every large polyhedral map on any compact 2-manifoldM with Euler characteristic (M)<0 contains a pathP _k such that each of its vertices has degree 6k – 2 ifk is odd,k3, (3) moreover, these bounds are attained. Fork=1 ork even,k2, the bound is 6k which has been proved in our previous paper.     

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.     

Staircase kernels for orthogonald-polytopes

LetS be a finite union of boxes inR ^d . Forx inS, defineA _x ={yx is clearly visible fromy via staircase paths inS}, and let KerS denote the staircase kernel ofS. Then KerS={A _x x is a point of local nonconvexity ofS}. A similar result holds with clearly visible replaced by visible and points of local nonconvexity ofS replaced by boundary points ofS.Supported in part by NSF grant DMS-9207019.     

On the number of nowhere zero points in linear mappings

LetA be a nonsingularn byn matrix over the finite fieldGF _q ,k=n/2,q=p ^a ,a1, wherep is prime. LetP(A,q) denote the number of vectorsx in (GF _q ) ⁿ such that bothx andAx have no zero component. We prove that forn2, and ,P(A,q)[(q–1)(q–3)] ^k (q–2) ^n–2k and describe all matricesA for which the equality holds. We also prove that the result conjectured in [1], namely thatP(A,q)1, is true for allqn+23 orqn+14.     

Maximum antichains in the product of chains

Let P be the poset k ₁ × ... × k _n , which is a product of chains, where n1 and k ₁ ... k _n 2. Let . P is known to have the Sperner property, which means that its maximum ranks are maximum antichains. Here we prove that its maximum ranks are its only maximum antichains if and only if either n=1 or M1. This is a generalization of a classical result, Sperner's Theorem, which is the case k ₁= ... =k _n =2. We also determine the number and location of the maximum ranks of P.Research supported in part by the National Science Foundation 10/25/83.     

Arithmetic Properties of Summands of Partitions

Let d d, d2 2. We prove that for almost all partitions of an integer the parts are well distributed in residue classes mod d. The limitations of the uniformity of this distribution are also studied.     

The (-1/2λϕ2) model of quantum field theory

The model of a scalar field with interaction potential exp (-1/2²) ind-dimensional spacetime (d2) is considered. It is shown that the Green's functions of the model are the same as the Green's functions of the free field.Physicotechnical Institute, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 3, pp. 411–417, June, 1992.