  A subset of a metric space is a k-distance set if there are exactly k
non-zero distances occuring between points. We conjecture that a k-distance set
in a d-dimensional Banach space (or Minkowski space), contains at most (k+1)^d
points, with equality iff the unit ball is a parallelotope. We solve this
conjecture in the affirmative for all 2-dimensional spaces and for spaces where
the unit ball is a parallelotope. For general spaces we find various weaker
upper bounds for k-distance sets.
