  The Falconer conjecture asserts that if E is a planar set with Hausdorff
dimension strictly greater than 1, then its Euclidean distance set has positive
one-dimensional Lebesgue measure. We discuss the analogous question with the
Euclidean distance replaced by non-Euclidean norms in which the unit ball is a
polygon with 2K sides. We prove that for any such norm there is a set of
Hausdorff dimension K/(K-1) whose distance set has Lebesgue measure 0.
