On the location of geometrical medians of triangles

封面

如何引用文章

全文:

开放存取 开放存取
受限制的访问 ##reader.subscriptionAccessGranted##
受限制的访问 订阅存取

详细

The geometrical median is a natural spatial generalization of the statistical median of a one-dimensional sample. Thus the problem of computing the median of a finite set of points (a sample) on a straight line presents no difficulties, but unexpected difficulties arise in moving to the plane or to higher dimensional spaces, where the natural linear order of points is absent. While the mean of a multidimensional sample, as on a straight line, is calculated by taking the arithmetic mean, no such analytical formula is available for the geometric median. Moreover, such formulas are absent when we deal with geometrical medians of continuous objects located on a plane or in space. This raises the natural question of analytical estimates of the locations of geometric medians. This paper presents the solutions for two such simplest problems. Namely, the solution of the problem on estimating the location of the geometric median of the perimeter of a triangle and the solution of a similar problem on the geometric median of a triangular area. For both problems, we obtain exact estimates of the affine type.

全文:

受限制的访问

作者简介

P. Panov

HSE University

编辑信件的主要联系方式.
Email: ppanov@hse.ru
俄罗斯联邦, Moscow

参考

  1. Балк М. Б., Болтянский В. Г. (1987). Геометрия масс. М.: Наука. [Balk M. B., Boltyansky V. G. (1987). Geometry of masses. M.: Nauka (in Russian).]
  2. Панов П. А. (2017). Равновесные расположения центров благ по городу // Журнал Новой экономической ассоциации. № 1. С. 28–42. [Panov P. A. (2017). Nash equilibria in the facility location problem with externalities. Journal of the New Economic Association, 1 (33), 28–42 (in Russian).]
  3. Панов П. А. (2021). О геометрических медианах треугольников. Режим доступа: https://arxiv.org/ftp/arxiv/papers/2007/2007.14231.pdf [Panov P. A. (2021). On geometric medians of triangles. Available at: https://arxiv.org/ftp/arxiv/papers/2007/2007.14231.pdf (in Russian).]
  4. Bajaj C. (1988). The algebraic degree of geometric optimization problems. Discrete and Computational Geometry, 3 (2), 177–191.
  5. Behrend K. (2014). Introduction to algebraic stacks. In: Moduli Spaces. L. Brambila-Paz, P. Newstead, R. P. Thomas, O. García-Prada (eds.). London Mathematical Society Lecture Notes, 411. Cambridge: Cambridge Univ. Press., 1–131.
  6. Fekete S. P., Mitchell J. S.B., Beurer K. (2005). On the continuous Fermat-Weber problem. Operations Research, 53 (1), 61– 76. doi: 10.1287/opre.1040.0137. S2CID1121
  7. Mallows C. (1991). Another comment on O’Cinneide. The American Statistician, 45, 3, 257. doi: 10.1080/00031305.1991.10475815
  8. Murray A. T. (2020). Location theory. In: International encyclopedia of human geography. 2nd ed. A. Kobayashi (ed.). Oxford: Elsevier. doi: 10.1016/B978-0-08-102295-5.10104-0
  9. Piché R. (2012). Random vectors and random sequences. Saarbrücken: Lambert Academic Publishing. ISBN: 978-3659211966
  10. Stewart I. (2017). Why do all triangles form a triangle? American Mathematical Monthly, 124, 1, 70–73. doi: 10.4169/amer.math.monthly.124.1.70
  11. Yao J., Zhang X., Murray A. T. (2019). Location optimization of urban fire stations: Access and service coverage. Computers, Environment and Urban Systems, 73, 184–190. doi: 10.1016/j.compenvurbsys.2018.10.006

补充文件

附件文件
动作
1. JATS XML
2. Fig. 1. The geometric median m1(δ) is contained inside a small triangle δ′ similar to δ, and the geometric median m2(δ) is contained inside a small curvilinear triangle δ″

下载 (65KB)
3. Fig. 2. Degenerate triangle, a = b + c

下载 (10KB)
4. Fig. 3. On the left in the triangle is the display image - the region, on the right is an enlarged copy

下载 (127KB)
5. Fig. 4. Triangle (left - region - display image, right - enlarged copy)

下载 (80KB)

版权所有 © Russian Academy of Sciences, 2024