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.     

Families of planar sets having starlike union

Let be a finite family of compact sets in the plane, and letk be a fixed natural number. If every three (not necessarily distinct) members of have a union which is simply connected and starshaped viak-paths, then and is starshaped viak-paths. Analogous results hold for paths of length at most , > 0, and for staircase paths, although not for staircasek-paths.Supported in part by NSF grant DMS-9504249     

Some Krasnosel'skii numbers for finitely starlike sets in the plane

Let S be a subset of the plane. In case (int cl S) S = , then S is finitely starlike if and only if every 4 points of S see via S a common point. In case (int cl S) S has at most countably many components, each a singleton set, then S is finitely starlike if and only if every 5 points of S see via S a common point. Each of the numbers 4 and 5 is best possible. Examples show that these results fail without suitable restrictions on (int cl S) S. Moreover, a final example shows that if a general Krasnosel'skii number . exists to characterize finitely starlike sets in the plane, then > 9.     

Determining starshaped sets and unions of starshaped sets by their sections

Let S be a compact set in R^d. Let p be a fixed point of S and let k be a fixed integer, 1 k <d. Then S is starshaped with p ker S if and only if for every k-dimensional flat F through p, F S is starshaped. Moreover, an analogue of this result holds for unions of starshaped sets as well.     

Points of local nonconvexity,clear visibility,and starshaped sets in Rd

Let S be a compact, connected, locally starshaped set in R^d, S not convex. For every point of local nonconvexity q of S, define A_q to be the subset of S from which q is clearly visible via S. Then ker S = {conv A_q: q lnc S}. Furthermore, if every d+1 points of local nonconvexity of S are clearly visible from a common d-dimensional subset of S, then dim ker S = d.     

Points of local nonconvexity and sets which are almost starshaped

Let Sø be a bounded connected set in R ², and assume that every 3 or fewer lnc points of S are clearly visible from a common point of S. Then for some point p in S, the set A{s : s in S and [p, s] S} is nowhere dense in S. Furthermore, when S is open, then S in starshaped.     

Illumination by translates of convex sets

Let S be a compact set in the plane. If every three points of S are illuminated clearly by some translate of the compact convex set T, then there is a translate of T which illumines every point of S. Various analogues hold for translates of flats in R ^das well.Supported in part by NSF grant DMS-8705336.     

Unions of three starshaped sets in R2

LetS be a compact, simply connected set inR ². If every boundary point ofS is clearly visible viaS from at least one of the three pointsa, b, c, thenS is a union of three starshaped sets whose kernels containa, b, c, respectively. The result fails when the number three is replaced by four.As a partial converse, ifS is a union of three starshaped sets whose kernels containa, b, c, respectively, then the set of points in the boundary ofS clearly visible from at least one ofa, b, orc is dense in the boundary ofS.Supported in part by NSF grant DMS-8705336.     

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.     

Simply connected orthogonal polygons as unions of two orthogonally starshaped sets

Let S be a simply connected orthogonal polygon in the plane. The set S is a union of two sets which are starshaped via staircase paths (i.e., orthogonally starshaped) if and only if for every three points of S, at least two of these points see (via staircase paths) a common point of S. Moreover, the simple connectedness condition cannot be deleted.     

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     

A Helly theorem for intersections of orthogonally starshaped sets

Let $\cal{F}$ be a finite family of simply connected orthogonal polygons in the plane. If every three (not necessarily distinct) members of $\cal{F}$ have a nonempty intersection which is starshaped via staircase paths, then the intersection $\cap \{F : F\; \hbox{in}\; \cal{F}\}$ is a (nonempty) simply connected orthogonal polygon which is starshaped via staircase paths. Moreover, the number three is best possible, even with the additional requirement that the intersection in question be nonempty. The result fails without the simple connectedness condition.     

On coverings of plane convex sets by translates of strips

In the euclidean planeE ² letS ₁,S ₂, ... be a sequence of strips of widthsw ₁,w ₂, .... It is shown thatE ² can be covered by translates of the stripsS _i if w ₁ ^3/2 = . Further results concern conditions in order that a compact convex domain inE ² can be covered by translates ofS ₁,S ₂, ....This research was supported by National Science Foundation Research Grant MCS 76-06111.     

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.     

The dimension of the kernel in finite and infinite intersections of starshaped sets

Summary. We establish the following Helly-type result for infinite families of starshaped sets in Define the function f on {1, 2} by f(1) = 4, f(2) = 3. Let be a fixed positive number, and let be a uniformly bounded family of compact sets in the plane. For k = 1, 2, if every f(k) (not necessarily distinct) members of intersect in a starshaped set whose kernel contains a k-dimensional neighborhood of radius , then is a starshaped set whose kernel is at least k-dimensional. The number f(k) is best in each case. In addition, we present a few results concerning the dimension of the kernel in an intersection of starshaped sets in Some of these involve finite families of sets, while others involve infinite families and make use of the Hausdorff metric.     

A problem of Kusner on equilateral sets

R. B. Kusner [R. Guy, Amer. Math. Monthly 90, 196-199 (1983)] asked whether a set of vectors in such that the distance between any pair is 1, has cardinality at most d + 1. We show that this is true for p = 4 and any , and false for all 1<p<2 with d sufficiently large, depending on p. More generally we show that the maximum cardinality is at most if p is an even integer, and at least if 1<p<2, where depends on p. Received: 5 May 2003     

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Dem Wirken des hervorragenden ungarischen GeometersWolfgang Bolyai zum 200. Geburtstag