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This paper examines the complexity of several geometric problems due to unbounded dimension. The problems considered are: (i) minimum cover of points by unit cubes, (ii) minimum cover of points by unit balls, and (iii) minimum number of lines to hit a set of balls. Each of these problems is proven not to have a polynomial approximation scheme unless P = NP. Specific lower bounds on the error ratios attainable in polynomial time are given, assuming P NP. In particular, it is shown that covering by two cubes is in P while covering by three cubes is NP-complete.