Isosceles set

The unique 6-point isosceles set in the plane. The shaded regions show four of the 20 isosceles triangles formed by triples of these points.

In discrete geometry, an isosceles set is a set of points with the property that every three of them form an isosceles triangle. More precisely, each three points should determine at most two distances; this also allows degenerate isosceles triangles formed by three equally-spaced points on a line.

The problem of finding the largest isosceles set in a Euclidean space of a given dimension was posed in 1946 by Paul Erdős. In his statement of the problem, Erdős observed that the largest such set in the Euclidean plane has six points.[1] In his 1947 solution, Leroy Milton Kelly showed more strongly that the unique six-point planar isosceles set consists of the vertices and center of a regular pentagon. In three dimensions, Kelly found an eight-point isosceles set, six points of which are the same; the remaining two points lie on a line perpendicular to the pentagon through its center, at the same distance as the pentagon vertices from the center.[2] This three-dimensional example was later proven to be optimal, and to be the unique optimal solution.[3][4]

In -dimensional space, an isosceles set can have at most

points.[5] This is tight for and for but not necessarily for other dimensions. The maximum number of points in a -dimensional isosceles set, for , is known to be[6]

3, 6, 8, 11, 17, 28, 30, 45 (sequence A175769 in the OEIS)

but these numbers are not known for higher dimensions.[7]

The same problem can also be considered for other metric spaces. For instance, for Hamming spaces, somewhat smaller upper bounds are known than for Euclidean spaces of the same dimension.[7] In an ultrametric space, the whole space (and any of its subsets) is an isosceles set. Therefore, ultrametric spaces are sometimes called isosceles spaces. However, not every isosceles set is ultrametric; for instance, obtuse Euclidean isosceles triangles are not ultrametric.[8]

References

  1. ↑ Grossman, Howard; Thebault, Victor; Schell, E. D.; Scheffe, Henry; Erdős, Paul (August 1946), "Problems for Solution: E731–E735", The American Mathematical Monthly, 53 (7): 394, doi:10.2307/2305860 . See in particular problem E735.
  2. ↑ Erdős, Paul; Kelly, L. M. (April 1947), "E735", The American Mathematical Monthly, 54 (4): 227, doi:10.2307/2304710
  3. ↑ Croft, H. T. (1962), "9-point and 7-point configurations in 3-space", Proceedings of the London Mathematical Society, Third Series, 12: 400–424, doi:10.1112/plms/s3-12.1.400, MR 0155230
  4. ↑ Kido, Hiroaki (2006), "Classification of isosceles eight-point sets in three-dimensional Euclidean space", Electronic Journal of Combinatorics, 27 (3): 329–341, doi:10.1016/j.ejc.2005.01.003, MR 2206471
  5. ↑ Blokhuis, A. (1984), Few-distance sets, CWI Tract, 7, Amsterdam: Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, MR 0751955
  6. ↑ Lisoněk, Petr (1997), "New maximal two-distance sets", Journal of Combinatorial Theory, Series A, 77 (2): 318–338, doi:10.1006/jcta.1997.2749, MR 1429084
  7. 1 2 Ionin, Yury J. (2009), "Isosceles sets", Electronic Journal of Combinatorics, 16 (1): Research Paper 141, 24, MR 2577309
  8. ↑ Fiedler, Miroslav (1998), "Ultrametric sets in Euclidean point spaces", Electronic Journal of Linear Algebra, 3: 23–30, doi:10.13001/1081-3810.1012, MR 1615350
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