j-partitions for visible shorelines

Summary LetC be a compact set inR ². A setS R ² C is said to have aj-partition relative toC if and only if there existj or fewer pointsc ₁,, c _j inC such that each point ofS sees somec _i via the complement ofC. Letm, j be fixed integers, 3 m, 2 j, and writem (uniquely) asm = qj + r, where 1 r j. Assume thatC is a convexm-gon in R², withS R ² C. Forq = 0 orq = 1, the setS has aj-partition relative toC. Forq 2,S has aj-partition relative toC if and only if every (qj + 1)-member subset ofS has aj-partition relative toC, and the Helly numberqj + 1 is best possible.IfC is a disk, no such Helly number exists.     

Unions of orthogonally convex or orthogonally starshaped polygons

LetT be a simply connected orthogonal polygon having the property that for every three points ofT, at least two of these points see each other via staircases inT. ThenT is a union of three orthogonally convex polygons. The number three is best possible.ForT a simply connected orthogonal polygon,T is a union of two orthogonally convex polygons if and only if for every sequencev ₁,...,v _n,v _n+1 =v ₁ inT, n odd, at least one consecutive pairv _i ,v _i+1 sees each other via staircase paths inT, 1 i n. An analogous result says thatT is a union of two orthogonal polygons which are starshaped via staircase paths if and only if for every odd sequence inT, at least one consecutive pair sees a common point via staircases inT.Supported in part by NSF grants DMS-8908717 and DMS-9207019.     

Z4-Kerdock Codes, Orthogonal Spreads, and Extremal Euclidean Line-Sets

When m is odd, spreads in an orthogonal vector space of type⁺(2m + 2,2) are related to binary Kerdock codes and extremalline-sets in 2^{m + 1} with prescribed angles. Spreads in a 2m-dimensionalbinary symplectic vector space are related to Kerdock codesover Z₄ and extremal line-sets in with prescribed angles. These connections involve binary, realand complex geometry associated with extraspecial 2-groups.A geometric map from symplectic to orthogonal spreads is shownto induce the Gray map from a corresponding Z₄-Kerdock codeto its binary image. These geometric considerations lead tothe construction, for any odd composite m, of large numbersof Z₄-Kerdock codes. They also produce new Z₄-linear Kerdockand Preparata codes. 1991 Mathematics Subject Classification:primary 94B60; secondary 51M15, 20C99.     

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.     

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.     

A characterization theorem for tilings having countably many singular points

LetT be a tiling of the plane. At most countably many points of U{bdry T T inT} fail to lie in a nondegenerate edge ofT if and only ifT has at most countably many singular points. The result fails without the requirement that the edges be nondegenerate. Moreover, countably many cannot be replaced by finitely many in the theorem.     

An improved Krasnosel'skii theorem for nonclosed sets in the plane

Let S be a nonempty bounded set in R² whose complement consists of a finite number of locally compact components. Assume that every 3 or fewer points in S see a common point via S. Then for some point p in cl S, the set A ≡ {s ∶ s in S and (p,s]? S} is nowhere dense in S. The number 3 is best possible.