Let E(n) be n-dimensional Euclidean space. It is known that there exist subsets of E(n) which are simple arcs and have positive n-dimensional Lebesgue measure when n=1 or 2. Does this continue to be true for arbitrarily large n? If not, what is the largest n for which it holds and is there a simple proof of this fact? Intuitively, I feel that there should be no upper bound, but cannot see how to prove it.