Preserving entanglement under perturbation and sandwiching all separable states

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Abstract
Every entangled state can be perturbed and stay entangled. For a large class of pure entangled states, which includes all bipartite and all maximally entangled ones, we show how large the perturbation can be. Maximally entangled states can be perturbed the most. For each entangled state in our class, we construct hyperplanes which sandwich the set of all separable states. As the number of particles, or the dimensions of the Hilbert spaces for two of the particles increases, the distance between two of these hyperplanes goes to zero.