Balancing Unit Vectors

Theorem A.Letx1, …, x2k+1be unit vectors in a normed plane. Then there exist signsε1, …, ε2k+1∈{±1} such that ‖∑2k+1i=1 εixi‖⩽1. 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). Theorem B.Letp0, p1, …, pnbe distinct points in a normed plane such that for any 1⩽i<j⩽nthe closed angle ∠pi p0 pjcontains a ray opposite some[formula], 1⩽k⩽n. 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. Theorem C.Letx1, x2, … be a sequence of unit vectors in a normed plane. Then there exist signsε1, ε2, …∈{±1} such that ‖∑2ki=1 εixi‖⩽2 for allk∈N. Finally we discuss a variation of a two-player balancing game of J. Spencer (1977) related to Theorem C.