This question is related to https://mathoverflow.net/questions/50600/an-existence-question-on-linear-map. If the answer to this question is yes, it would solve the abovementioned other MO question.

We equip ${\mathbb R}^3$ with the $\ell_3$ norm $||(x,y,z)||=(|x|^3+|y|^3+|z|^3)^{\frac{1}{3}}$. Is it true that, for any vectorial plane $V$ in ${\mathbb R}^3$, we can find a vector $w$ not in $V$ such that $||w+v|| \geq ||v||$ for all $v \in V$ ? It is easily seen that this property holds for some norms (such as the $\ell_2$ norm) and fails for others, such as the $\ell_{\infty}$ norm.