Since my intuition for high dimensional geometry is not always right:


Consider the unit cube in $\mathbb{R}^m$ and for $n\leq m$ denote by $F^n$ the union of the $n$-facets. For what numbers of $m$ and $n$ does any $n$-dimensional subspace of $\mathbb{R}^m$ intersect $F^n$?


Extra: Consider the same question for $G^n$ = $F^n\setminus F^{n-1}$.


(P.S: What kind geometry tag would be appropriate?)



Does an $n$ dimensional subspace intersect the $n$-facets of the unit cube?
The condition for the first question is: $2n \geq m.$


