Sets in \(E^ 3\) that locally lie on flat spheres (Q1077763)

The \(\epsilon\)-boundary, \(\partial (\epsilon,X)\), of a subset X of Euclidean 3-space \(E^ 3\) is the set \(\{\) \(p|\) \(d(p,X)=\epsilon \}\) where d is the usual metric for \(E^ 3\). The main theorem states that a subset A of \(E^ 3\) locally lies on a flat 2-spheres at each of its points if A is the \(\epsilon\)-boundary of a subset X of \(E^ 3\) for some \(\epsilon >0\) and if \(E^ 3=A\cup N(X,\epsilon)\) where \(N(X,\epsilon)=\{p|\) \(d(X,p)<\epsilon \}\). From this it follows that one-dimensional sets locally lie on flat 2-spheres when they are realized as an \(\epsilon\)-boundary. Previously it had been proved that \(\partial (\epsilon,X)\) was locally flat at each point where it was known to be a 2-manifold and that \(\partial (\epsilon,X)\) was a 2-manifold for almost all \(\epsilon\).