Abstract
We consider a variant of Heilbronn’s triangle problem by asking for fixed integers d,k ≥2 and any integer n ≥k for a distribution of n points in the d-dimensional unit cube [0,1]d such that the minimum volume of the convex hull of k points among these n points is as large as possible. We show that there exists a configuration of n points in [0,1]d, such that, simultaneously for j = 2, ..., k, the volume of the convex hull of any j points among these n points is Ω( 1/n (j − − 1)/(1 + |d − − j + 1|)). Moreover, for fixed k ≥d+1 we provide a deterministic polynomial time algorithm, which finds for any integer n ≥k a configuration of n points in [0,1]d, which achieves, simultaneously for j = d+1, ..., k, the lower bound Ω( 1/n (j − − 1)/(1 + |d − − j + 1|)) on the minimum volume of the convex hull of any j among the n points.
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Ajtai, M., Komlós, J., Pintz, J., Spencer, J., Szemerédi, E.: Extremal Uncrowded Hypergraphs. Journal of Combinatorial Theory Ser. A 32, 321–335 (1982)
Barequet, G.: A Lower Bound for Heilbronn’s Triangle Problem in d Dimensions. SIAM Journal on Discrete Mathematics 14, 230–236 (2001)
Barequet, G.: The On-Line Heilbronn’s Triangle Problem in Three and Four Dimensions. In: H. Ibarra, O., Zhang, L. (eds.) COCOON 2002. LNCS, vol. 2387, pp. 360–369. Springer, Heidelberg (2002)
Bertram-Kretzberg, C., Hofmeister, T., Lefmann, H.: An Algorithm for Heilbronn’s Problem. SIAM Journal on Computing 30, 383–390 (2000)
Brass, P.: An Upper Bound for the d-Dimensional Heilbronn Triangle Problem. SIAM Journal on Discrete Mathematics 19, 192–195 (2005)
Cassels, J.W.S.: An Introduction to the Geometry of Numbers, vol. 99. Springer, New York (1971)
Chazelle, B.: Lower Bounds on the Complexity of Polytope Range Searching. Journal of the American Mathematical Society 2, 637–666 (1989)
Jiang, T., Li, M., Vitany, P.: The Average Case Area of Heilbronn-type Triangles. Random Structures & Algorithms 20, 206–219 (2002)
Komlós, J., Pintz, J., Szemerédi, E.: On Heilbronn’s Triangle Problem. Journal of the London Mathematical Society 24, 385–396 (1981)
Komlós, J., Pintz, J., Szemerédi, E.: A Lower Bound for Heilbronn’s Problem. Journal of the London Mathematical Society 25, 13–24 (1982)
Lefmann, H.: On Heilbronn’s Problem in Higher Dimension. Combinatorica 23, 669–680 (2003)
Lefmann, H.: Large Triangles in the d-Dimensional Unit-Cube. In: Chwa, K.-Y., Munro, J.I.J. (eds.) COCOON 2004. LNCS, vol. 3106, pp. 43–52. Springer, Heidelberg (2004)
Lefmann, H.: Distributions of Points in the Unit-Square and Large k-Gons. In: Proceedings ACM-SIAM Syposium on Discrete Algorithms, SODA 2005, pp. 241–250. ACM/SIAM (2005)
Lefmann, H.: Large Simplices in the d-Dimensional Unit-Cube (Extended Abstract). In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 514–523. Springer, Heidelberg (2005)
Lefmann, H., Schmitt, N.: A Deterministic Polynomial Time Algorithm for Heilbronn’s Problem in Three Dimensions. SIAM Journal on Computing 31, 1926–1947 (2002)
Roth, K.F.: On a Problem of Heilbronn. Journal of the London Mathematical Society 26, 198–204 (1951)
Roth, K.F.: On a Problem of Heilbronn, II, and III. Proc. of the London Mathematical Society 25(3), 193–212, 543–549 (1972)
Roth, K.F.: Estimation of the Area of the Smallest Triangle Obtained by Selecting Three out of n Points in a Disc of Unit Area. In: Proc. of Symposia in Pure Mathematics, AMS, Providence, vol. 24, pp. 251–262 (1973)
Roth, K.F.: Developments in Heilbronn’s Triangle Problem. Advances in Mathematics 22, 364–385 (1976)
Schmidt, W.M.: On a Problem of Heilbronn. Journal of the London Mathematical Society 4(2), 545–550 (1972)
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Lefmann, H. (2006). Distributions of Points and Large Convex Hulls of k Points. In: Cheng, SW., Poon, C.K. (eds) Algorithmic Aspects in Information and Management. AAIM 2006. Lecture Notes in Computer Science, vol 4041. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11775096_17
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DOI: https://doi.org/10.1007/11775096_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-35157-3
Online ISBN: 978-3-540-35158-0
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