Intersection of a cube and a line in $R^n$.

Question

Assuming in $R^n$ we have a line and a cube with nonempty intesection. Is the itersection a line? This true for $n=1,2,3$. But, is this true for any $n$? Can anybody recommend me a good reference book about this topic? Thank you so much!

Answer 1

Hint: a cube $Q$ is always a convex subset of $\mathbb{R}^n:$ you can convince about this considering coordinates $x_1,\dots,x_n$ such that $Q=\{x\in\mathbb{R}^n|a_1\leq x_1\leq a_1+l, \dots, a_n\leq x_n\leq a_n+l\}$ where $l$ is the size of the edge and $(a_1,\dots,a_n)\in \mathbb{R}^n.$

Thanks to convexity it is easy to deduce that the intersection is a segment, which can be intended also in its degenerate case, that is a single point.

Ps: be careful then, because the intersection is not a line, which has infinite length, but a segment, that is a portion of a line.