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.
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