Finding the supremum of a 4D epsilon neighborhood

[bobbarker]

Homework Statement

Find sup{[tex]\epsilon[/tex]| N_{[tex]\epsilon[/tex]}(X₀ [tex]\subset[/tex] S} for
X₀ = (1,2,-1,3); S = open 4-ball of radius 7 about (0,3,-2,2).

Homework Equations

If X₁ is in S_r(X₀) and
|X - X₁| < [tex]\epsilon[/tex] = r - |X - X₀|
then X is in S_r(X₀)

The Attempt at a Solution

This is my first foray into n-dimensional analysis and I'm pretty intimidated. I understand the theory behind it I think--we're making a little n-dimensional ball around the point X₀, and in fact trying to maximize the size of the ball while still remaining in the bigger ball our set is defined by.

I'm confused about the actual determination of the size of [tex]\epsilon[/tex] though. What do I use for this arbitrary point X in the equation?

This is a multi-part problem and I could do the ones in dimensions I could visualize. :P Can anyone help me out with this 4D stuff? Thanks.

Edit: the answer in the back of the book says the supremum is 5.

 

[Mark44]

You have a ball of radius 7 around the point (0, 3, -2, 2).

How big a ball can you have around (1, 2, -1, 3) so that it fits inside the ball of radius 7?

A sketch of the situation in 2D will be helpful, even despite the fact that you're dealing with a four-dimension space.