Question

Show that for any set V consisting of $n\ge 1$ vectors, the number of maximal subsets is less than or equal to $2n$

Answer 1

Let us draw the vectors of V as originated from the same point O. Consider any maximal subset $B\subset V$, and denote by u the sum of all vectors from B. If l is the line through O perpendicular to u, then B contains exactly those vectors from V that lie on the same side of l as u does, and no others. Indeed, if any $v\text{/}ϵB$ lies on the same side of l, then $|u+v|\ge |u|$; similarly, if some $vϵB$ lies on the other side of l. then $|u-v|\ge |u|$.
Therefore every maximal subset is determined by some line l as the set of vectors lying on the same side of l. It is obvious that in this way we get at most $2n$ sets.