A norm - inequality related to affine regular hexagons

Let $(E, \lVert . \rVert)$ be a two-dimensional real normed space with unit sphere $S = \{x \in E, \lVert x \rVert = 1\}$. The main result of this paper is the following: Consider an affine regular hexagon with vertex set $H = \{\pm v_1, \pm v_2, \pm v_3\} \subseteq S$ inscribed to $S$. Then we have $$\min_i \max_{x \in S}{\lVert x - v_i \rVert + \lVert x + v_i \rVert} \leq 3.$$ From this result we obtain $$\min_{y \in S} \max_{x \in S}{\lVert x - y \rVert + \lVert x + y \rVert} \leq 3,$$ and equality if and only if $S$ is a parallelogram or an affine regular hexagon.