A krasnosel'skii-type theorem in the plane involving polygonal n-paths

It is shown that thenth order kernel of a compact simply connected subsetS ofR ² is nonempty if and only if every three boundary points ofS are visible via polygonaln-paths from a common point inS.     

Dimensions of staircase kernels in orthogonal polygons

LetS be a simply connected orthogonal polygon in the plane. If every four points ofS see (via staircase paths inS) a common segment, then the staircase kernel ofS contains a segment as well. A parallel result holds when every four points ofS see (via staircase paths inS) a common 2-dimensional set. In each case, the number 4 is best possible. However, both results fail without the requirement that setS be simply connected.An example shows that no Helly-type analogue is possible for intersections of orthogonally convex sets in the plane.     

A theorem on level lines of continuous functions

A property of a continuous functionf(x), x ∈ E ₂, similar to the classical intermediate value property is established. Namely, let a Jordan compactJ ⊂ E ₂ be the domain of definition off. Then, for each parametrizationx(t), 0≦t≦T,x(0)=x(T), of the boundary Fr(J) ofJ there exists a unique real λ and a unique connected component Φ of the level set {x ∈ J: f(x)=λ} with the following property: any connected subset Ω ofJ containing “opposite” points of Fr(J) (i.e. pointsx(t′) andx(t″) such thatt″−t′=T/2) has a common element with Φ.     

On sets having finitely many points of local nonconvexity and propertyP m

If a pointq ofS has the property that each neighborhood ofq contains pointsx andy such that the segmentxy is not contained byS, q is called a point of local nonconvexity ofS. LetQ denote the set of points of local nonconvexity ofS. Tietze’s well known theorem that a closed connected setS in a linear topological space is convex ifQ=φ is generalized in the result:If S is a closed set in a linear topological space such that S ∼ Q is connected and |Q|=n<∞,then S is the union of n+1or fewer closed convex sets. Letk be the minimal number of convex sets needed in a convex covering ofS. Bounds fork in terms ofm andn are obtained for sets having propertyP _m and |Q|=n.     

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.     

A Krasnosel’skii-type theorem for paths of bounded length

Let S be a nonempty closed, simply connected set in the plane, and let α τ; 0. If every three points of 5 see a common point of S via paths of length at most α, then for some point s0 of S, s0 sees each point of S via such a path. That is, S is starshaped via paths of length at most α. Supported in part by NSF grant DMS-9207019     

Convexity and a certain propertyP m

The propertyP _m (directly analogous to Valentine’s propertyP ₃) is used to prove several curious results concerning subsets of a topological linear space, among them the following: (a) If a closed setS has propertyP _m and containsk points of local nonconvexity no distinct pair of which can see each other viaS, thenS is the union ofm − k − 1 or fewer starshaped sets. (b) Any closed connected set with propertyP _m is polygonally connected. (c) A closed connected setS with propertyP _m is anL _m−1 set (each pair of points may be joined by a polygonal arc ofm − 1 of fewer sides inS). (d) A finite-dimensional set with propertyP _m is anL _{2m − 3} set. A new proof of Tietze’s theorem on locally convex sets is given, and various examples refute certain plausible conjectures.     

AnR danalogue of Valentine’s theorem on 3-convex sets

This paper deals with anR ^danalogue of a theorem of Valentine which states that a closed 3-convex setS in the plane is decomposable into 3 or fewer closed convex sets. In Valentine’s proof, the points of local nonconvexity ofS are treated as vertices of a polygonP contained in the kernel ofS, yielding a decomposition ofS into 2 or 3 convex sets, depending on whetherP has an even or odd number of edges. Thus the decomposition actually depends onc(P′), the chromatic number of the polytopeP′ dual toP. A natural analogue of this result is the following theorem: LetS be a closed subset ofR ^d, and letQ denote the set of points of local nonconvexity ofS. We require thatQ be contained in the kernel ofS and thatQ coincide with the set of points in the union of all the (d − 2)-dimensional faces of somed-dimensional polytopeP. ThenS is decomposable intoc(P′) closed convex sets.     

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.     

ON STABLE LINE SEGMENTS IN ALL TRIANGULATIONS OF A PLANAR POINT SET

Abstract. Let S be a set of finite plauar points. A llne segment L(p, q) with p, q E Sis called a stable line segment of S, if there is no Line segment with two endpoints in S intersecting L(p, q). In this paper, some geometric properties of the set of all stable line segments     

The subalgebra systems of direct powers

A simple characterization of the subalgebra systems of direct powers of finitary universal algebras on a fixed infinite setA is given. For |I|≥|A| such subalgebra system of anI-power is precisely an algebraic closure systemS onA ^I closed under mutations ofI (which encompass both the precomposition by permutations ofI and allowing the values at specified elements ofI to become unrestricted) and such that each function in the intersection ofS is constant. For |I|<|A| the subalgebra systems ofI-powers are obtained as the restrictions toI of such closure systems on someA ^J withJ⊇I and |J|=|A|. Presented by J. D. Monk.     

Spectrum,harmonic functions,and hyperbolic metric spaces

The main result of the paper says, in particular, that ifM is a complete simply connected Riemannian manifold with Ricci curvature bounded from below and without focal points, which is also a hyperbolic metric space in the sense of Gromov, then the top λ of theL ²-spectrum of the Laplace-Beltrami operator Δ is negative, the Martin boundary ofM corresponding to Δ is homeomorphic to the sphere at infinityS(∞), and the harmonic measures onS(∞) have positive Hausdorff dimensions. These generalize the results of [AS], [An1], [Ki], [KL] and [BK]. Moreover, if dimM=2, then in the presence of the other conditions the hyperbolicity is also necessary for λ<0. The machinery consists of a combination of geometrical and probabilistic means. Partially supported by U.S.-Israel BSF. Partially sponsored by the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

Staircase two-guard kernels of orthogonal polygons

Let S be a simply connected orthogonal polygon in the plane, and assume that S is two-guardable (but not starshaped) via staircase paths. If K is a component of two-kernel (S), then the set of partners of points in K determines a second component K' of two-kernel (S). Thus the components occur in pairs. Moreover, each component is geodesically convex. The results fail without the requirement that S be simply connected. Received 2 November 1999; revised 28 September 2000.     

Sequentially equidistant points in metric spaces

A sequence (z ₀,z ₁,z ₂,, ...,z _n, z _n+1) of points fromp=z ₀ toq=z _n+1 in a metric spaceX is said to besequentially equidistant ifd(z _i−1,z _i)=d(z _i,z _i+1) for 1≦i≦n. If there is path inX fromp toq (or if a certain weaker condition holds), then such a sequence exists, with all points distinct, for every choice ofn, while ifX is compact and connected, then such a sequence exists at least forn=2. An example is given of a dense connected subspaceS ofR ^m ,m≧2, and an uncountable dense subsetE disjoint fromS for which there is no sequentially equidistant sequence of distinct points (n ≧ 2) inS ∪E between any two points ofE. Techniques of dimension theory are utilized in the construction of these examples, as well as in the proofs of some of the positive results. Supported in part by NSF Grant DMS-8701666.     

On the structure of pseudo-holomorphic discs with totally real boundary conditions

In this paper, we study the structure ofJ-holomorphic discs in relation to the Fredholm theory of pseudo-holomorphic discs with totally real boundary conditions in almost complex manifolds (M, J). We prove that anyJ-holomorphic disc with totally real boundary condition that is injective in the interior except at a discrete set of points, which we call a “normalized disc,” must either have some boundary point that is regular and has multiplicity one, or satisfy that its image forms a smooth immersed compact surface (without boundary) with a finite number of self-intersections and a finite number of branch points. In the course of proving this theorem, we also prove several theorems on the local structure of boundary points ofJ-holomorphic discs, and as an application we give a complete treatment of the transverslity result for Floer’s pseudo-holomorphic trajectories for Lagrangian intersections in symplectic geometry. This paper is supported in part by NSF Grant DMS 9215011.     

Remarks on Gelfand Pairs Associated to Non-Type-I Solvable Lie Groups

LetSbe a connected and simply connected unimodular solvable Lie group andKa connected compact Lie group acting onSas automorphisms. We call the pair (K S) a Gelfand pair if the Banach ∗-algebraL¹_K(S) of allK-invariant integrable functions onSis a commutative algebra. In this paper we give a necessary and sufficient condition for the pair (K; S) to be a Gelfand pair using the representation theory of non-type-I solvable Lie groups. For a Gelfand pair (K; S) we realize all irreducibleK-spherical representations ofK?Sfrom irreducible unitary representations ofS.     

On the intersection of maximalm-convex subsets

A subsetS of a real linear spaceE is said to bem-convex providedm≧2, there exist more thanm points inS, and for eachm distinct points ofS at least one of the ( ₂ ^m ) segments between thesem points is included inS. InE, letxy denote the segment between two pointsx andy. For any pointx inSυE, letS _x ={y: xyυS}. The kernel of a setS is then defined as {xεS: S _x=S}. It is shown that the kernel of a setS is always a subset of the intersection of all maximalm-convex subsets ofS. A sufficient condition is given for the intersection of all the maximalm-convex subsets of a setS to be the kernel ofS.     

On support points for some families of univalent functions

Given a closed subset of the familyS* (α) of functions starlike of order α, a continuous Fréchet differentiable functional,J, is constructed with this collection as the solution set to the extremal problem ReJ(f) overS* (α). The support points ofS* (α) is completely characterized and shown to coincide with the extreme points of its convex hulls. Given any finite collection of support points ofS* (α), a continuous linear functional,J, is constructed with this collection as the solution set to the extremal problem ReJ(f) overS* (α).     

Guarding galleries where every point sees a large area

We prove a conjecture of Kavraki, Latombe, Motwani and Raghavan that ifX is a compact simply connected set in the plane of Lebesgue measure 1, such that any pointx∈X sees a part ofX of measure at least ɛ, then one can choose a setG of at mostconst1/ɛ log 1/ɛ points inX such that any point ofX is seen by some point ofG. More generally, if for anyk points inX there is a point seeing at least 3 of them, then all points ofX can be seen from at mostO(k ³ logk) points. Research supported by grants from the Sloan Foundation, the Israeli Academy of Sciences and Humanities, and by G.I.F. Research supported by Czech Republic Grant GAČR 201/94/2167 and Charles University grants No. 351 and 361. Part of the work was done while the author was visiting The Hebrew University of Jerusalem.     

A decomposition theorem form-convex sets

LetS be a closedm-convex subset of the plane,m≧2,Q the set of points of local nonconvexity ofS, with convQ ⊆S. If there is some pointp in [(bdryS) ∩ (kerS)] ∼Q, thenS is a union ofm−1 closed convex sets. The result is best possible for everym.