Geometry in $\mathbb{R}^n$: angle between projections of a rectangle

Question

Consider a hyper rectangle $R$ in $\mathbb{R}^n$ defined by $|x_i|\leq M_i$ for all $i\leq n$. Consider a linear affine subspace $L$ of dimension $1\leq k <n$ such that $L\cap R\neq \emptyset$. For all $x \in R\setminus L$, we define the point $x'\in L\cap R$ that minimizes the euclidian distance of $x$ to $L\cap R$. We also denote by $x_{\perp}$ the orthogonal projection of $x$ on $L$. If $\alpha_x$ is the angle between the vectors $x'-x$ and $x_{\perp}-x$, I would like to prove that $\inf_{x\in R\setminus L} |\cos(\alpha_x)|>0$. Looking at dimension 2, it seems to me that this is a reasonable conjecture but I was not able to prove it. Can someone give a help please?

Answer 1

Let $\ell$ be a perpendicular line to $L$. We can assume that $\ell$ does not lie in a coordinate hyperplane (otherwise, the question can be reduced to a lower-dimensional case). Denote by $\alpha$ the maximal angle between $\ell$ and the coordinate lines; note that $\alpha<\tfrac\pi2$.

Look at the tangent cone at $x'$. Note that $\alpha_x$ is the angle between $\ell$ and $x-x'$. Observe that there is a supporting hyperplane $\Pi$ to $R$ thru $x'$ such that $(x_\perp-x' )\perp(L\cap \Pi)$. Moreover, we may assume that $\Pi$ contains an edge of $R$. Denote by $\beta$ the angle between $\Pi$ and $\ell$. Note that $\alpha\geqslant \beta\geqslant \alpha_x$ --- hence your statement.