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The sign-sequence constant of the plane

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 $v_1, \ldots, v_n \in B$, there are signs $\varepsilon_1, \ldots, \varepsilon_n \in \{-1, +1\}$ such that $\varepsilon_1 v_1 + \ldots + \varepsilon_k v_k \in t B$, for each $1 \leq k \leq n$. 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 $\sqrt{3}$.