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 $${\sqrt{3}}$$ .