Abstract
The Fink-Wood problem on the contractibility of half-spaces of partial convexity is studied. It is proved that there exists a connected non-simply-connected half-space of orthocon-vexity in the three-dimensional space, which disproves the Fink-Wood conjecture in the general case. In a special case, it is proved that, if the set of directions of partial convexity contains a basis of the linear n-dimensional space, then all directed half-spaces of partial convexity are contractible.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. G. Naidenko, “Partial convexity,” Mat. Zametki 75(2), 222–235 (2004) [Math. Notes 75 (1–2), 202–212 (2004)].
V. G. Naidenko, “On a version of partial convexity,” Vestsī Nats. Akad. Navuk Belarusī Ser. Fīz.-Mat. Navuk no. 2, 102–106 (2004) [in Russian].
V.G. Naidenko, “Construction of OC-convex hulls,” Vestsī Nats. Akad. Navuk Belarusī Ser. Fīz.-Mat.Navuk no. 4, 48–53 (2002) [in Russian].
N. N. Metel’skii and V. G. Naidenko, “Algorithmic aspects of partial convexity,” Mat. Zametki 68(3), 399–410 (2000) [Math. Notes 68 (3–4), 345–354 (2000)].
E. Fink and D. Wood, Restricted-Orientation Convexity, Monogr. Theoret. Comput. Sci. EATCS Ser. (Springer-Verlag, Berlin, 2004).
B. Grünbaum, Convex Polytopes (Interscience Publ., New York, 1967).
V. V. Gorokhovik and E. A. Semenkova, “Classification of half-spaces by their types in infinite-dimensional vector spaces,” Mat. Zametki 64(2), 191–198 (1998) [Math. Notes 64 (1–2), 164–169 (1998), (1999)].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V. G. Naidenko, 2009, published in Matematicheskie Zametki, 2009, Vol. 85, No. 6, pp. 915–926.
Rights and permissions
About this article
Cite this article
Naidenko, V.G. Contractibility of half-spaces of partial convexity. Math Notes 85, 868–876 (2009). https://doi.org/10.1134/S0001434609050277
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434609050277