The sign-sequence constant of the plane

Abstract

Let $L$ be a finite-dimensional real normed space, and let $B$ be the unit ball in $L$. The sign sequence constant of $L$ is the least $t>0$ such that, for each sequence $v_1,\dots ,v_n\in B$, there are signs ${\epsilon }_1,\dots ,{\epsilon }_n\in \{-1,+1\}$ such that ${\epsilon }_1{v}_1+\dots +{\epsilon }_k{v}_k\in tB$, for each $1\le k\le n$. We show that the sign sequence constant of a plane is at most $2$, and the sign sequence constant of the plane with the Euclidean norm is equal to $\sqrt{3}$.

7 pages