I've been thinking about this. Suppose you have an n-point set P in R(adsbygoogle = window.adsbygoogle || []).push({}); ^{m}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 R^{m}that contains P, allowing for both translations and orthogonal transformations of S?

If we start in R^{2}, 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 = (x_{1}, x_{2}) : ||x - (0,1)|| < 2, x_{1}< 0, x_{2}< 0} U {x = (x_{2},x_{2}) : x_{1}= 0, -1 < x_{2}< 1}

Any thoughts?

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# Smallest Set Containing All n-point Sets

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