Consider some set of $d$ linearly independent unit vectors $x_1,\cdots,x_d$ in $\mathbb{R}^d$. Let $\mathcal{H}$ denote the unique hyperplane passing through all of these points. Is there any relation between the smallest singular value, $\sigma_d$ of the $d \times d$ matrix $(x_1 \ x_2 \ \cdots \ x_d)$ and the Euclidean distance $d(0,\mathcal{H})$ from the origin to $\mathcal{H}$?
Perhaps there is, since $\sigma_d = 0 \iff d(0,\mathcal{H}) = 0$.
$\sigma_d = 0$ means that the matrix $X$ has a non-trivial nullspace and hence $\exists v$ such that for every $i$, $\langle x_i,v \rangle = 0$. Therefore $\mathcal{H}$ is the plane orthogonal to $v$ and passing through the origin. The reverse implication follows suit.