A norm - inequality related to affine regular hexagons

Abstract

Let $(E,\Vert .\Vert )$ be a two-dimensional real normed space with unit sphere $S=\{x\in E,\Vert x\Vert =1\}$. 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 $$\underset{i}{\min }\underset{x\in S}{\max }\Vert x-v_i\Vert +\Vert x+v_i\Vert \le 3.$$ From this result we obtain $$\underset{y\in S}{\min }\underset{x\in S}{\max }\Vert x-y\Vert +\Vert x+y\Vert \le 3,$$ and equality if and only if $S$ is a parallelogram or an affine regular hexagon.