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Gaussian random projections for Euclidean membership problems

We discuss the application of random projections to the fundamental problem of deciding whether a given point in a Euclidean space belongs to a given set. We show that, under a number of different assumptions, the feasibility and infeasibility of this problem are preserved with high probability when the problem data is projected to a lower dimensional space. Our results are applicable to any algorithmic setting which needs to solve Euclidean membership problems in a high-dimensional space.