Abstract
Suppose we have two convex sets A and B in euclidean d-space ℝd. Assume the only information we have about A and B comes from the space of their transversal lines. Can we determine whether A and B have a point in common? For example, suppose the space of their transversal lines has an essential curve; that is, suppose there is a line that moves continuously in ℝd, always remaining transversal to A and B, and comes back to itself with the opposite orientation. If this is so, then A must intersect B, otherwise there would be a hyperplane H separating A from B; but it turns out that our moving line becomes parallel to H at some point on its trip, which is a contradiction to the fact that the moving line remains transversal to the two sets. If we have three convex sets A, B and C, for example, in ℝ3, then our essential curve does not give us sufficient topological information. In this case, to detect whether A ∩ B ∩ C ≠ ϕ, we need a 2-dimensional cycle. So, for example, if we can continuously choose a transversal line parallel to every direction, then there must be a point in A ∩ B ∩ C, otherwise if not, the same is true for π(A)∩π(B)∩π(C), for a suitable orthogonal projection π: ℝ3 → H where H is a plane through the origin (see [4, Lemma 3.1]). Hence clearly there is no transversal line orthogonal to H.
Supported by CONACYT, 41340.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
J. Arocha, J. Bracho, L. Montejano, D. Oliveros and R. Strausz, Separoids; their categories and a Hadwiger-type theorem for transversals. Journal of Discrete and Computational Geometry, Vol. 27, No. 3 (2002), 377–385.
J. Arocha, J. Bracho, L. Montejano and J. Ramirez-Alfonsin, Transversals to the convex hull of all k-sets of discrete subsets of ℝn (2009) preprint.
I. Bárány and S. Onn, Carathéodory’s Theorem, colorful and applicable. In Intuitive Geometry (Budapest, 1995) volume 6 of Bolyai Soc. Math. Stud., pp. 11–21. János Bolyai Math. Soc., Budapest, 1997.
J. Bracho and L. Montejano, Helly type theorems on the homology of the space of transversals. Journal of Discrete and Computational Geometry, Vol. 27, No. 3 (2002), 387–393.
J. Bracho, L. Montejano and D. Oliveros, The topology of the space of transversals through the space of configurations. Topology and Its Applications Vol. 120, No. 1–2 (2002), 92–103.
S. S. Chern, On the multiplication in the characteristic ring of a sphere bundle, Annals of Math., 49, (1948), 362–372.
J. E. Goodman, R. Pollack, and R. Wenger, Geometric transversal theory. In J. Pach, ed., New Trends in Discrete and Computational Geometry, vol. 10 of Algorithms Combin., pp. 163–198. Springer-Verlag, Berlin, 1995.
J. W. Milnor and J. D. Stasheff, Characteristic Classes. Annals of Mathematical Studies No. 76, Princeton University Press, N.J., 1974.
L. Montejano and R. Karasev, Topological Transversals to a Family of Convex Sets. To appear in Discrete and Computational Geometry, 2010.
L. Montejano and P. Soberon, Piercing numbers for balanced and unbalanced families. To appear in Discrete and Computational Geometry, 2010.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 János Bolyai Mathematical Society and Springer-Verlag
About this chapter
Cite this chapter
Montejano, L. (2013). Transversals, Topology and Colorful Geometric Results. In: Bárány, I., Böröczky, K.J., Tóth, G.F., Pach, J. (eds) Geometry — Intuitive, Discrete, and Convex. Bolyai Society Mathematical Studies, vol 24. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41498-5_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-41498-5_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-41497-8
Online ISBN: 978-3-642-41498-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)