ABSTRACT
Then, the level sets of u are hyperplanes, at least if N ≤ 8. Note that if the level sets of u are hyperplanes, then they must be parallel (no two level sets can cross). So the conclusion of the conjecture is equivalent to requesting that u is one-dimensional (that is, u is a function of one variable only; see Definition 6.2.1). One-dimensional solutions of (7.1) are easily classified, as the following proposition demonstrates.
