j-partitions for visible shorelines

Breen, Marilyn

doi:10.1007/bf01836091

Cited by 5 publications

(10 citation statements)

References 5 publications

(4 reference statements)

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“…, p(k), p(k + 1). The point s(i o ) must see a point of color c (1), and the only way this can occur is for p(k + 1) to be color c (1). Recall that r ≥ 1.…”

Section: Results For H(n Q R)mentioning

confidence: 99%

“…Then our listed set of points contains only s (1), no two of the s(i) points can be consecutive, and at least half the points of P must be skips. If n ≥ 2, this yields at least k+r 2 ≥ 2q+r 2 ≥ q skips, and the result is immediate.…”

Section: Results For H(n Q R)mentioning

confidence: 99%

“…It is convenient and a bit more intuitive to pass from the combinatorial problem above to an analogous geometric problem: We may think of P as the vertex set of a convex (qn+r)-gon C, think of S as a subset of R 2 \C, and, for each member s(i) of S, think of V (i) as the corresponding visibility set of s(i), the set of q vertices of C seen by s(i) via the complement of C. (See [7], [1], [2] for related definitions.) Then the earlier combinatorial problem may be rephrased in geometric terms similar to those used by Valentine in [7], with C an island and S a set of swimmers.…”

Section: Introductionmentioning

confidence: 99%

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Helly-type theorems for appropriate colorings of visibility sets

Breen

2002

Aequationes Mathematicae

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For positive integers q and n, think of P as the vertex set of a (qn + r)-gon, 0 ≤ r ≤ q − 1. For 1 ≤ i ≤ qn + r, define V (i) to be a set of q consecutive points of P , starting at p(i), and let S be a subset of {V (i) : 1 ≤ i ≤ qn + r}. A q-coloring of P = P (q) such that each member of S contains all q colors is called appropriate for S, and when 1 ≤ j ≤ q, the definition may be extended to suitable subsets P (j) of P . If for every 1 ≤ j ≤ q and every corresponding P (j), P (j) has a j-coloring appropriate for S, then we say P = P (q) has all colorings appropriate for S. With this terminology, the following Helly-type result is established: Set P = P (q) has all colorings appropriate for S if and only if for every (2n + 1)-member subset T of S, P has all colorings appropriate for T . The number 2n + 1 is best possible for every r ≥ 1. Intermediate results for q-colorings are obtained as well. Mathematics Subject Classification (2000). Primary 52A30, 52A35.

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“…, p(k), p(k + 1). The point s(i o ) must see a point of color c (1), and the only way this can occur is for p(k + 1) to be color c (1). Recall that r ≥ 1.…”

Section: Results For H(n Q R)mentioning

confidence: 99%

Section: Results For H(n Q R)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Helly-type theorems for appropriate colorings of visibility sets

Breen

2002

Aequationes Mathematicae

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“…Breen [36] proved (in terms of visibility of neighborhoods, which is equivalent to the illumination by points, as mentioned at the end of Section 5) that if S is a compact set of exterior points to a compact, convex body K ⊂ E d such that any d + 1 points from S illuminate a common boundary point of K, then all points of S illuminate a common boundary point of K (see also [35] for similar problems on the visibility of convex polygons).…”

Section: Miscellaneous Resultsmentioning

confidence: 99%

Combinatorial problems on the illumination of convex bodies

Martini

Soltan

1999

Aequationes Mathematicae

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This is a review of various problems and results on the illumination of convex bodies in the spirit of combinatorial geometry. The topics under review are: history of the Gohberg-Markus-Hadwiger problem on the minimum number of exterior sources illuminating a convex body, including the discussion of its equivalent forms like the minimum number of homothetic copies covering the body; generalization of this problem for the case of unbounded convex bodies; visibility and inner illumination of convex bodies; primitive illuminating systems for convex bodies; illumination and visibility of families of convex bodies; clouds formed by translates or homothetic copies of a convex body; miscellaneous results.Mathematics Subject Classification (1991). 52A37, 52A40.

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“…In [1], using the notion of a j-partition, j ≥ 2, an analogue of Valentine's result was obtained for unions of j starshaped sets…”

Section: Introductionmentioning

confidence: 87%

Visible shorelines containing at least k vertices

Breen

2000

Aequat. Math.

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Let C be a two-dimensional convex polygon in the plane, S ⊆ R 2 \ C, and let f be a function defined on the natural numbers by f (1) = 5, f (2) = 4, and f (k) = 3 for k ≥ 3. For k fixed, k ≥ 1, all points of S see at least k common vertices of C via the complement of S if and only if every f (k) points of S see at least k common vertices of C via the complement of S. For each k, the corresponding number f (k) is best possible. (2000). Primary 52A35, 52A30. Mathematics Subject Classification

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j-partitions for visible shorelines

Cited by 5 publications

References 5 publications

Helly-type theorems for appropriate colorings of visibility sets

Helly-type theorems for appropriate colorings of visibility sets

Combinatorial problems on the illumination of convex bodies

Visible shorelines containing at least k vertices

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