  We give improved upper bounds on the radius of the largest ball of separable
states of an m-qubit system around the maximally mixed state. The ratio between
the upper bound and the best known lower bound (Hildebrand, quant.ph/0601201)
thus shrinks to a constant c = \sqrt{34/27} ~ 1.122, as opposed to a term of
order \sqrt{m\log m} for the best upper bound known previously (Aubrun and
Szarek, quant.ph/0503221). We give concrete examples of separable states on the
boundary to entanglement which realize these upper bounds. As a by-product, we
compute the radii of the largest balls that fit into the projective tensor
product of four unit balls in R^3 and in the projective tensor product of an
arbitrary number of unit balls in R^n for n = 2,4,8.
