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  • 1
    Electronic Resource
    Electronic Resource
    KRASNOSEL'SKII-TYPE THEOREMS (1985)
    Oxford, UK : Blackwell Publishing Ltd
    Annals of the New York Academy of Sciences 440 (1985), S. 0 
    Details
    ISSN: 1749-6632
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Natural Sciences in General
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1749-6632.1985.tb14548.x
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  • 2
    Electronic Resource
    Electronic Resource
    A Helly-Type Theorem for Intersections of Compact Connected Sets in the Plane (1998)
    Springer
    Geometriae dedicata 71 (1998), S. 111-117 
    Details
    ISSN: 1572-9168
    Keywords: Molnár's theorem ; Helly-type theorems ; compact connected sets ; planar sets.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For a family C of nonempty compact sets in the plane, the following conditions are equivalent: (1) Every two (not necessarily distinct) members of C have a connected union and every three (not necessarily distinct) members of C have a simply connected union. (2) C is a family of simply connected sets such that every two (not necessarily distinct) members of C have a connected intersection and every three (not necessarily distinct) members of C have a nonempty intersection. If either set of conditions is satisfied, then ∩ { C : C in C } is nonempty, simply connected, and connected. Furthermore, if the collection C is finite, then it is also true that ∪ { C : C in C } is simply connected.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1005003212822
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  • 3
    Electronic Resource
    Electronic Resource
    Unions of orthogonally convex or orthogonally starshaped polygons (1994)
    Springer
    Geometriae dedicata 53 (1994), S. 49-56 
    Details
    ISSN: 1572-9168
    Keywords: Primary 52A30 ; 51A35
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01264043
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  • 4
    Electronic Resource
    Electronic Resource
    Dimensions of staircase kernels in orthogonal polygons (1994)
    Springer
    Geometriae dedicata 49 (1994), S. 323-333 
    Details
    ISSN: 1572-9168
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01264032
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  • 5
    Electronic Resource
    Electronic Resource
    ℒ2 sets which are almost starshaped (1977)
    Springer
    Geometriae dedicata 6 (1977), S. 485-494 
    Details
    ISSN: 1572-9168
    Keywords: Primary 52.A30 ; 52.A10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let S be a subset of R d . The set S is said to be an ℒ set if and only if for every two points x and y of S, there exists some z ∈ S such that [x, z] ⋃ [z, y] ⊂ S. Clearly every starshaped set is an ℒ set, yet the converse is false and introduces an interesting question: ‘Under what conditions will an ℒ set S be “almost” starshaped; that is, when will there exist a convex subset C of S such that every point of S sees some point of C via S’ This paper provides one answer to the question above, and we have the following result: Let S be a closed planar ℒ set, S simply connected, and assume that the set Q of points of local nonconvexity of S is finite. If some point p of S see each member of Q via S, then there is a convex subset C of S such that every point of S sees some point of C via S.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00147785
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  • 6
    Electronic Resource
    Electronic Resource
    A Helly-type theorem for simple polygons (1996)
    Springer
    Geometriae dedicata 60 (1996), S. 283-288 
    Details
    ISSN: 1572-9168
    Keywords: Primary 52A35 ; secondary 52A10 ; Helly theorems ; simple polygons ; orthogonally convex polygons
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let $$\mathcal{P}$$ be a family of simple polygons in the plane. If every three (not necessarily distinct) members of $$\mathcal{P}$$ have a simply connected union and every two members of $$\mathcal{P}$$ have a nonempty intersection, then ∩{P:P in $$\mathcal{P}$$ } ≠ $$\not 0$$ . Applying the result to a finite family $$\mathcal{C}$$ of orthogonally convex polygons, the set ∩{C:C in $$\mathcal{C}$$ } will be another orthogonally convex polygon, and, in certain circumstances, the dimension of this intersection can be determined.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00147365
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  • 7
    Electronic Resource
    Electronic Resource
    A quantitative Krasnosel'skii theorem in R d (1982)
    Springer
    Geometriae dedicata 12 (1982), S. 219-226 
    Details
    ISSN: 1572-9168
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00147642
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  • 8
    Electronic Resource
    Electronic Resource
    Points of local nonconvexity and sets which are almost starshaped (1982)
    Springer
    Geometriae dedicata 13 (1982), S. 201-213 
    Details
    ISSN: 1572-9168
    Keywords: Primary 52.A30 ; 52.A35 ; secondary 52.A10 ; Krasnosel' skii-type theorems ; starshaped sets ; convex kernel ; points of local nonconvexity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let S≠ø 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00147663
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  • 9
    Electronic Resource
    Electronic Resource
    Some tilings of the plane whose singular points are uncountable and unbounded (1984)
    Springer
    Geometriae dedicata 17 (1984), S. 139-144 
    Details
    ISSN: 1572-9168
    Keywords: Primary 52.A45
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract 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:n≥1} 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00151501
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  • 10
    Electronic Resource
    Electronic Resource
    A characterization theorem for certain unions of two starshaped sets in R 2 (1987)
    Springer
    Geometriae dedicata 22 (1987), S. 95-103 
    Details
    ISSN: 1572-9168
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let S be a compact set in R 2. For S simply connected, S is a union of two starshaped sets if and only if for every F finite, F $$ \subseteq $$ bdry S, there exist a set G $$ \subseteq $$ bdry S arbitrarily close to F and points s, t depending on G such that each point of G is clearly visible via S from one of s, t. In the case where ∼S has at most finitely many components, the necessity of the condition still holds while the sufficiency fails.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00183056
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