- #1
jgm340
- 104
- 2
I've been thinking about this. Suppose you have an n-point set P in Rm which has the property that for any two points x, y in P, ||x - y|| < 2. If we fix n, what can we say about the smallest set S in Rm that contains P, allowing for both translations and orthogonal transformations of S?
If we start in R2, the answer is not what you'd expect! As illustrated in this picture, P must be able to fit inside of any 2 by 2 square after only translation:
That is, if I were to have a square hoola-hoop, I could swing it around P. However, for n > 2, the intersection of all those squares would not necessarily fit inside a circle of diameter 2.
For n=2, the smallest set would be something like [0,1] U {2}.
For n=3, I think the smallest set would be {x = (x1, x2) : ||x - (0,1)|| < 2, x1 < 0, x2 < 0} U {x = (x2,x2) : x1 = 0, -1 < x2 < 1}
Any thoughts?
If we start in R2, the answer is not what you'd expect! As illustrated in this picture, P must be able to fit inside of any 2 by 2 square after only translation:
That is, if I were to have a square hoola-hoop, I could swing it around P. However, for n > 2, the intersection of all those squares would not necessarily fit inside a circle of diameter 2.
For n=2, the smallest set would be something like [0,1] U {2}.
For n=3, I think the smallest set would be {x = (x1, x2) : ||x - (0,1)|| < 2, x1 < 0, x2 < 0} U {x = (x2,x2) : x1 = 0, -1 < x2 < 1}
Any thoughts?