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Visible shoreline results for staircase paths in ℝd
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Marilyn Breen
Published/Copyright:
January 17, 2014
Abstract. Let S be a staircase convex orthogonal polytope in ℝd. For every d and j; 1 ≤ d and 0 ≤ j ≤ d - 1, let ƒ(d; j) denote the smallest number n such that the following holds: For P ⊆ ℝd\(int S), if every n points of P see a common j-dimensional subset of bdry S via staircase paths in ℝd\(int S), then all points of P see a common j-dimensional subset of bdry S via such paths. When S belongs to a certain class of orthogonal polytopes called elementary, it follows that
• ƒ(d; d - 1) = 2d for d ≥ 1,
• ƒ(d; j) ≤ 2d for d ≥ 2; 0 ≤ j ≤ d - 2,
• ƒ(2; j) = 4 for 0 ≤ j ≤ 1, and
• ƒ(3; j) = 6 for 0 ≤ j ≤ 2.
References
Published Online: 2014-01-17
Published in Print: 2014-01
© 2014 by Walter de Gruyter GmbH & Co.
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Articles in the same Issue
- Masthead
- Real hypersurfaces of complex and quaternionic hyperbolic spaces
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- Invariant surfaces in Sol3 with constant mean curvature and their computer graphics
- A short geometric proof that Hausdorff limits are definable in any o-minimal structure
- Seshadri constants on smooth threefolds
- Visible shoreline results for staircase paths in ℝd
- Characterization of isometric embeddings of Grassmann graphs
- Typical simplicially convex bodies
- On successive radii of p-sums of convex bodies
- Resolving sets for four families of distance-regular graphs
- White-black graphs and right-angled hyperbolic handlebodies
- Scalar and Ricci curvatures of special contact slant submanifolds in Sasakian space forms
- Non-factorial quartic double solids
- Collineation groups of topological parallelisms
