  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.
