Median hyperplanes in normed spaces - A survey

Discrete Applied Mathematics

Volumn 89, Issue 1-3, 1998, Pages 181-195

  Martini, Horst ^a   Schöbel, Anita ^b  

^a CHEMNITZ UNIVERSITY OF TECHNOLOGY   (Germany)

^b University of Koblenz Landau   (Germany)

Author keywords

Block norms; Continuous location; Facility location; Halving hyperplanes; Linear fit problem; Linear L1 approximation; Median hyperplane; Minkowski space; Minsum hyperplane; Orthogonal L1 regression; Path location; Planar k sets

Indexed keywords

EID: 0013053816     PISSN: 0166218X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0166-218X(98)00103-6     Document Type: Article

References (54)

1. B. Aronov, B. Chazelle, H. Edelsbrunner, L. Guibas, M. Sharir, R. Wenger, Points and triangles in the plane and halving planes in space, Discrete Comput. Geometry 6 (1991) 435-442.
2. I. Bárány, Z. Füredi, L. Lovász, On the number of halving planes, Combinatorica 10 (1990) 175-183.
3. R.J. Boscovich, De litteraria expeditione per pontificiam ditionem, et synopsis amplioris operis, ac habentur plura ejus ex exemplaria etiam sensorum impressa, Bononiensi Scientiarum et Artum Instituto atque Academia Commentarii 4 (1757) 353-396.
4. T.K. Dey, H. Edelsbrunner, Counting triangle crossings and halving planes, Discrete Comput. Geometry 12 (1994) 281-289.
5. F.Y. Edgeworth, On observations relating to several quantities, Hermathena 6(13) (1887) 279-285.
6. F.Y. Edgeworth, On a new method of reducing observations relating to several quantities, Phil. Magazine (Series 5) 25 (1888) 184-191.
7. H. Edelsbrunner, L.J. Guibas, Topologically sweeping an arrangement, J. Comput. Systems Sci. 38 (1989) 165-194.
8. H. Edelsbrunner, J. O'Rourke, R. Seidel, Constructing arrangements of lines and hyperplanes with applications, SIAM J. Comput. 15 (1986) 341-363.
9. P. Erdös, L. Lovász, A. Simmons, E. Strauss, Dissection graphs of planar point sets, in: J. Srivastara el al. (Eds.), A Survey of Combinatorial Theory, 1973, pp. 139-149.
10. W.D. Fisher, A note on curve fitting with minimum deviations by linear programming, J. Amer. Statist. Assoc. 56 (1961) 359-362.
11. H.W. Hamacher, Mathematische Lösungsverfahren für planare Standortprobleme, Vieweg, Braunschweig, 1995.
12. H.W. Hamacher, S. Nickel, Multicriteria planar location problems, Eur. J. Oper. Res. 94 (1996) 66-86.
13. ∞-approximation and applications. Lecture Notes Computer Science 382 (1989) 183-191.
14. ∞-approximation and applications, Discrete Applied Mathematics 43 (1993) 217-232.
15. N.M. Korneenko, Optimal Lines in the Plane, Math. Research 51 (1989) 43-51.
16. N.M. Korneenko, H. Martini, Approximating finite weighted point sets by hyperplanes, Lecture Notes Computer Science 447 (1990) 276-286.
17. N.M. Korneenko, H. Martini, Hyperplane approximation and related topics, in: J. Pach (Ed.), New Trends in Discrete and Computational Geometry, Ch. 6, Springer, New York, 1993, pp. 135-162.
18. D.T. Lee, Geometric location problems and their complexity. Lecture Notes Computer Science 233 (1986) 154-167.
19. D.T. Lee, Y.T. Ching, The power of geometric duality revisited, Inform. Process. Letters 21 (1985) 117-122.
20. D.T. Lee, Y.F. Wu, Geometric Complexity of Some Location Problems, Algorithmica 1 (1986) 193-211.
21. L. Lovász, On the number of halving lines, in: Ann. Univ. Eötvös Loránd, Sekt. Math., vol. 14, Budapest, 1971, pp. 107-108.
22. R.F. Love, J.G. Morris, G.O. Wesolowsky, Facilities Location, Ch. 3.3, North-Holland, Amsterdam, 1988, pp. 51-60.
23. H. Martini, Some results and problems around zonotopes, in: K. Böröczky, G. Fejes Tóth (Eds.), Intuitive Geometry, Coll. Math. Soc. J. Bolyai, vol. 48, North-Holland, 1987, pp. 383-418.
24. H. Martini, A. Schöbel, The minisum hyperplane problem in Minkowski spaces, Technical Report, University of Kaiserslautern, Wirtschaftsmathematik, 1997, submitted.
25. H. Martini, A. Schöbel, A characterization of smooth norms, Technical Report, 1998, in preparation.
26. N. Megiddo, A. Tamir, On the complexity of locating linear facilities in the plane, Oper. Res. Lett. 1(5) (1982) 194-197.
27. N. Megiddo, A. Tamir, Finding least-distance lines, SIAM J. Algebraic Discrete Methods 4(2) (1983) 207-211.
28. J.A. Mesa, Continuous location of dimensional structures, Working paper, presented at the Workshop on Locational Analysis, 1995.
29. H. Minkowski, Gesammelte Abhandlungen, Band 2, Chelsea, New York, 1967.
30. J.G. Morris, J.P. Norback, A simple approach to linear facility location, Transportation Science 14(1) (1980) 1-8.
31. J.G. Morris, J.P. Norback, Linear Facility Location - Solving extensions of the basic problem, Eur. J. Oper. Res. 12 (1983) 90-94.
32. S.C. Narula, J.F. Wellington, The minimum sum of absolute errors regression: a state of the art survey, Internat. Statist. Rev. 50 (1982) 317-326.
33. J.P. Norback, J.G. Morris, Fitting hyperplanes by minimizing orthogonal deviations, Math. Programming 19 (1980) 102-105.
34. M.H. Overmars, J. van Leeuwen, Dynamically maintaining configurations in the plane, J. Comput. Syst. Sci. 23 (1981) 166-204.
35. J. Pach, W. Steiger, E. Szemeredi, An upper bound on the number of planar k-sets, Discrete Comput. Geometry 7 (1992) 109-123.
36. F. Plastria, Continuous Location Problems, in: Z. Drezner (Ed.), Facility Location: A Survey of Applications and Methods, Ch. 11, Springer, New York, 1995, pp. 225-262.
37. W.H. Press, B.P. Flannery, S.A. Teulosky, W.T. Vetterling, Numerical Recipes, Cambridge University Press, Cambridge, New York, 1986.
38. M.R. Rao, V. Srinivasan, A note on Sharpe's algorithm for minimizing the sum of absolute deviations in a simple regression problem, Management Sci. 19 (1972) 222-225.
39. E.M. Reingold, J. Nievergelt, N. Deo, Combinatorial Algorithms: Theory and Practice, Prentice-Hall, Englewood Cliffs, New Jersey, 1977.
40. J. Rice, The Approximation of Functions: The Linear Theory, vol. 1, Addison-Wesley, Reading, MA, 1964,
41. P.J. Rousseeuw, A.M. Leroy, Robust Regression and Outlier Detection, Wiley, New York, 1987.
42. E.J. Schlossmacher, An iterative technique for absolute deviations curve fitting, J. Amer. Statist. Assoc. 68 (1973) 857-859.
43. A. Schöbel, Locating least-distant lines with block norms, Studies in Locational Analysis 10 (1996) 139-150.
44. A. Schöbel, Locating least distant lines in the plane, Eur. J. Oper. Res. 106 (1998) 139-152.
45. A. Schöbel, Locating line segments with vertical distances, Studies in Locational Analysis 11 (1997), 143-158.
46. M.I. Shamos, Computational Geometry, PhD, Thesis, Department of Computer Science, Yale University, New Haven, 1978.
47. 1 approximation, Numer. Math. 51 (1987) 531-543.
48. A.C. Thompson, Minkowski Geometry, Cambridge University Press, Cambridge, 1996.
49. Vrećica, R. Zivaljević, New cases of the colored Tverberg's theorem, in: Jerusalem Combinatorics '93, vol. 178 of Contemporary Mathematics (1994) pp. 325-334.
50. H.M. Wagner, Linear programming techniques for regression analysis, J. Amer. Statist. Assoc. 54 (1959) 206-212.
51. G.O. Wesolowsky, Location of the median line for weighted points, Environment and Planning A 7 (1975) 163-170.
52. 1 linear approximation of points, Proc. 4th Ann. Sympos. Comput. Geom., 1988, pp. 352-361.
53. E. Zemel, An O(n) algorithm for the linear multiple choice knapsack problem and related problems, Inform. Process. Lett. 18 (1984) 123-128.
54. R. Zivaljević, S. Vrećica, The colored Tverberg's problem and complexes of injective functions, J. Combin. Theory Series A 61 (1992) 309-318.