Balancing unit vectors

A. Letx1, …, x2k+1be unit vectors in a normed plane. Then there exist signs1, …, 2k+1{±1} such that ∑2k+1i=1 ixi1. We use the method of proof of the above theorem to show the following point facility location result, generalizing Proposition 6.4 of Y. S. Kupitz and H. Martini (1997). B. Letp0, p1, …, pnbe distinct points in a normed plane such that for any 1i<jnthe closed angle pi p0 pjcontains a ray opposite some[formula], 1kn. Thenp0is a Fermat–Torricelli point of {p0, p1, …, pn}, i.e.x=p0minimizes ∑ni=0 x−pi. We also prove the following dynamic version of Theorem A. C. Letx1, x2, … be a sequence of unit vectors in a normed plane. Then there exist signs1, 2, …{±1} such that ∑2ki=1 ixi2 for allkN. Finally we discuss a variation of a two-player balancing game of J. Spencer (1977) related to Theorem C.