Abstract
LetS≠⊘ be a finite union of boxes inR d. Forx inS, defineA x ={y∶x is clearly visible fromy via staircase paths inS}, and let KerS denote the staircase kernel ofS. Then KerS=▸{A x ∶x is a point of local nonconvexity ofS}. A similar result holds with “clearly visible” replaced by “visible” and “points of local nonconvexity ofS” replaced by “boundary points ofS”.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Breen, M.: A Krasnosel'skii theorem for staircase paths in orthogonal polygons. J. of Geometry51, 22–30 (1994).
Breen, M.: A Krasnosel'skii-type theorem for points of local nonconvexity. Proc. Amer. Math. Soc.85, 261–266 (1982).
Breen, M.: Points of local nonconvexity, clear visibility, and starshaped sets inR d. J. of Geometry21, 45–52 (1983).
Breen, M.: Staircase kernels in orthogonal polygons. Arch. Math.59, 588–594 (1992).
Danzer, L., Grünbaum B., Klee, V.: Helly's theorem and its relatives. In: Klee, V. L. (ed) Convexity. pp. 101–180. Proc. Sympos. Pure Math, Vol. 7, Providence, RI: Amer. Math. Soc. 1962.
Krasnosel'skii, M. A.: Sur un critère pour qu'un domainse soit étoilé. Math Sb. (61)19, 309–310 (1946).
Lay, S. R.: Convex Sets and Their Applications. New York: Wiley. 1982.
Motwani, R., Raghunathan, A., Saran, H.: Covering orthogonal polygons with starpolygons: the perfect graph approach. J. Comp. System Sci.40, 19–48 (1990).
O'Rourke, J.: Art Gallery Theorems and Algoriths, New York: Oxford Univ. Press. 1987.
Tietze, H.: Über Konvexheit im kleinen und im grossen und über gewisse den Punkten einer Menge zugeordnete Dimensionszahlen. Math. Z.28, 697–707 (1928).
Valentine, F. A.: Convex Sets. New York: McGraw-Hill. 1964.
Author information
Authors and Affiliations
Additional information
Supported in part by NSF grant DMS-9207019.
Rights and permissions
About this article
Cite this article
Breen, M. Staircase kernels for orthogonald-polytopes. Monatshefte für Mathematik 122, 1–7 (1996). https://doi.org/10.1007/BF01298451
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01298451