Abstract
Let \({\mathcal{S}}\) be a set system of convex sets in ℝd. Helly’s theorem states that if all sets in \({\mathcal{S}}\) have empty intersection, then there is a subset \({\mathcal{S}}'\subset{\mathcal{S}}\) of size d+1 which also has empty intersection. The conclusion fails, of course, if the sets in \({\mathcal{S}}\) are not convex or if \({\mathcal{S}}\) does not have empty intersection. Nevertheless, in this work we present Helly-type theorems relevant to these cases with the aid of a new pair of operations, affine-invariant contraction, and expansion of convex sets.
These operations generalize the simple scaling of centrally symmetric sets. The operations are continuous, i.e., for small ε>0, the contraction C −ε and the expansion C ε are close (in the Hausdorff distance) to C. We obtain two results. The first extends Helly’s theorem to the case of set systems with nonempty intersection:
(a) If \({\mathcal{S}}\) is any family of convex sets in ℝd, then there is a finite subfamily \({\mathcal{S}}'\subseteq{\mathcal{S}}\) whose cardinality depends only on ε and d, such that \(\bigcap_{C\in{\mathcal{S}}'}C^{-\varepsilon}\subseteq\bigcap_{C\in {\mathcal{S}}}C\) .
The second result allows the sets in \({\mathcal{S}}\) a limited type of nonconvexity:
(b) If \({\mathcal{S}}\) is a family of sets in ℝd, each of which is the union of k fat convex sets, then there is a finite subfamily \({\mathcal{S}}'\subseteq{\mathcal{S}}\) whose cardinality depends only on ε, d, and k, such that \(\bigcap_{C\in{\mathcal{S}}'}C^{-\varepsilon }\subseteq \bigcap_{C\in{\mathcal{S}}}C\) .
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References
Alon, N., Kalai, G.: Bounding the piercing number. Discrete Comput. Geom. 13(3/4), 245–256 (1995)
Amenta, N.: A short proof of an interesting Helly-type theorem. Discrete Comput. Geom. 15(4), 423–427 (1996)
Bárány, I., Pach, M.K.J.: Quantitative Helly-type theorems. Proc. Am. Math. Soc. 86(1), 109–114 (1982)
Barvinok, A.: A Course in Convexity. Graduate Studies in Mathematics, vol. 54. Am. Math. Soc., Providence (2002)
Demouth, J., Devillers, O., Glisse, M., Goaoc, X.: Helly-type theorems for approximate covering. In: Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry, pp. 120–128 (2008)
Dudley, R.M.: Metric entropy of some classes of sets with differentiable boundaries. J. Approx. Theory 10(3), 227–236 (1974)
Eckhoff, J.: Helly, Radon, and Carathéodory type theorems. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, vol. A, pp. 389–448. North-Holland, Amsterdam (1993)
Efrat, A., Katz, M.J., Nielsen, F., Sharir, M.: Dynamic data structures for fat objects and their applications. Comput. Geom. 15(4), 215–227 (2000)
Gao, J., Langberg, M., Schulman, L.J.: Clustering lines: classification of incomplete data. Manuscript (2006)
Gruber, P.M.: Aspects of approximation of convex bodies. In: Handbook of Convex Geometry, vol. A, pp. 319–345. North-Holland, Amsterdam (1993)
Grunbaum, B., Motzkin, T.S.: On components in some families of sets. Proc. Am. Math. Soc. 12(4), 607–613 (1961)
Helly, E.: Über Mengen konvexer Körper mit gemeinschaftlichen Punkten. Jahresber. Dtsch. Math.-Ver. 32, 175–176 (1923)
John, F.: Extremum problems with inequalities as subsidiary conditions. In: Studies and Essays Presented to R. Courant on his 60th Birthday, pp. 187–204. Interscience, New York (1948)
Kalai, G., Meshulam, R.: Leray numbers of projections and a topological Helly-type theorem. J. Topol. 1(3), 551–556 (2008)
Larman, D.G.: Helly type properties of unions of convex sets. Mathematika 15, 53–59 (1968)
Matoušek, J.: A Helly-type theorem for unions of convex sets. Comput. Geom. 18(1), 1–12 (1997)
Morris, H.: Two pigeonhole principles and unions of convexly disjoint sets. Ph.D. Thesis, Calif. Inst. of Techn., Calif. (1973)
Rademacher, H., Schoenberg, I.J.: Helly’s theorem on convex domains and Tchebycheff’s approximation problem. Can. J. Math. 2, 245–256 (1950)
Shnirelman, L.G.: On uniform approximations. Izv. Akad. Nauk SSSR Ser. Mat. 2, 53–60 (1938)
van der Stappen, A.F., Halperin, D., Overmars, M.H.: The complexity of the free space for a robot moving amidst fat obstacles. Comput. Geom. Theory Appl. 3, 353–373 (1993)
Wenger, R.: Helly-type theorems and geometric transversals. In: Handbook of Discrete and Computational Geometry, pp. 63–82. CRC Press, Boca Raton (1997)
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Research supported in part by grants from the NSF and the NSA.
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Langberg, M., Schulman, L.J. Contraction and Expansion of Convex Sets. Discrete Comput Geom 42, 594–614 (2009). https://doi.org/10.1007/s00454-009-9214-y
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DOI: https://doi.org/10.1007/s00454-009-9214-y