SPHERE INTERSECTIONS IN Eⁿ (n≥1)

   K = H(P₁,a₁) ∩ H(P₂,a₂) ∩ ... ∩ H(P_n,a_n)

               where

        a_i ≥ a_i+1 > 0        for i = 1, 2, ... , n-1.

               and

        P₁ ∈ Eⁿ  and

        P_i ∈ H(P₁,a₁) ∩ H(P₂,a₂) ∩ ... ∩ H(P_i-1,a_i-1)
                                   for i = 2, 3, ... , n.

DEFINITIONS

     • Eⁿ denotes Euclidean n-space. The set of n-tuples of
             real numbers.

     • If P ∈ Eⁿ and r is a real number greater than zero, then
                      H(P,r) = { Q ∈ Eⁿ | δ(P,Q) = r }.
       A "sphere" with center P and radius r.

     • δ(P,Q) denotes the distance between points P and Q in Eⁿ.

QUESTION

     Is K empty ?