- #1
TPAINE
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Homework Statement
Let x,y be in R^k, |x-y|=d. Prove there are an infinite number of solutions to |z-x|=|z-y|=r when 2r>d.
Homework Equations
I have done this problem for 2r=d and 2r<d. For 2r<d, the triangle inequality works nicely. For 2r=d, the equality gives me additional information to solve it. However, just setting |z-x|=|z-y|and expanding just get me a mess.
The Attempt at a Solution
The solution is obvious geometrically (the two "balls" intersect in a "circle") but I'd like to avoid undefined topological notions and just do it with the information in Rudin.