BORSUK COVERING AND PLANAR SETS WITH UNIQUE COMPLETION

The famous problem of Borsuk, whether every bounded set in R(n) can be covered by n + 1 sets of smaller diameter, is still open for n greate r-than-or-equal-to 4. We give an equivalent formulation of the problem . In the plane, the only sets which cannot be covered by two sets of s maller diameter are those whose completion is unique. We present a new characterization of planar sets with unique completion.