  The basic question that is addressed in this paper is finding the closest
separable state for a given entangled state, measured with the Hilbert Schmidt
distance. While this problem is in general very hard, we show that the
following strongly related problem can be solved: find the Hilbert Schmidt
distance of an entangled state to the set of all partially transposed states.
We prove that this latter distance can be expressed as a function of the
negative eigenvalues of the partial transpose of the entangled state, and show
how it is related to the distance of a state to the set of positive partially
transposed states (PPT-states). We illustrate this by calculating the closest
biseparable state to the W-state, and give a simple and very general proof for
the fact that the set of W-type states is not of measure zero. Next we show
that all surfaces with states whose partial transposes have constant minimal
negative eigenvalue are similar to the boundary of PPT states. We illustrate
this with some examples on bipartite qubit states, where contours of constant
negativity are plotted on two-dimensional intersections of the complete state
space.
