Abstract
A set X of boundary points of a (possibly unbounded) convex body K⊂E d illuminating K from within is called primitive if no proper subset of X still illuminates K from within. We prove that for such a primitive set X of an unbounded, convex set K⊂E d (distinct from a cone) one has ∣X∣∣∣=2 if d=2, ∣∣∣X∣∣∣≤6 if d=3, and that there is no upper bound for ∣X∣∣∣ if d≥4.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Boltyanski, V., Martini, H. and Soltan, P.: Excursions into Combinatorial Geometry, Springer-Verlag, Berlin, 1997.
Boltyanski, V., Martini, H. and Soltan, V.: On Grünbaum's conjecture about inner illumination of convex bodies, Discrete Comput. Geom., to appear.
Gale, D.: Neighborly and cyclic polytopes, In: Convexity (Seattle, 1961), Proc. Sympos. Pure Math. 7, Amer. Math. Soc., Providence, R.I., 1963, pp. 225–232.
Grünbaum, B.: Fixing systems and inner illumination, Acta Math. Acad. Sci. Hungar. 15 (1964), 161–163.
Grünbaum, B.: Convex Polytopes, Interscience Publishers, London, 1967.
Klee, V.: Extremal structure of convex sets, Arch. Math. (Basel) 8 (1957), 234–240.
Martini, H. and Soltan, V.: Combinatorial problems on the illumination of convex bodies, Aequationes Math. 57 (1999), 121–152.
Soltan, P.: On the illumination of the boundary of a convex body from within (Russian), Mat. Sb. 57 (1962), 443–448.
Soltan, P.: Illumination from within for unbounded convex bodies, (Russian) Dokl. Akad. Nauk SSSR 194 (1970), 273–274. (English translation: Soviet Math. Dokl. 11 (1970), 1199–1201.)
Soltan, V.: On Grünbaum's problem about inner illumination of convex bodies, Acta Math. Hungar. 69 (1995), 15–25.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Martini, H., Soltan, V. Primitive Inner Illuminating Systems for Convex Bodies. Geometriae Dedicata 80, 81–97 (2000). https://doi.org/10.1023/A:1005240410520
Issue Date:
DOI: https://doi.org/10.1023/A:1005240410520