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