“Isosceles set”的意思、由来-开放百科全书

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.{{r|e735}} 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.{{r|kelly}} This three-dimensional example was later proven to be optimal, and to be the unique optimal solution.{{r|croft|kido}}

points.{{r|blokhuis}} 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{{r|lisonek}}

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.{{r|ionin}} 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.{{r|fiedler}}

References

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