MathDB

Let

v_1, \ldots, v_{12}

be unit vectors in

\mathbb{R}^3

from the origin to the vertices of a regular icosahedron. Show that for every vector

v \in \mathbb{R}^3

and every

\varepsilon>0

, there exist integers

a_1, \ldots, a_{12}

such that

\left\|a_1 v_1+\cdots+a_{12} v_{12}-v\right\|<\varepsilon

Problem Statement