$\DeclareMathOperator{\complex}{\mathbb{C}}$ Let $X,Y \subseteq \complex^n$ be homogeneous, smooth, irreducible, closed algebraic sets with $X \cap Y=\{0\}$. I would like to numerically lower bound the quantity
\begin{align} \inf_{\substack{x\in X, y \in Y\\ ||x||_2=1,||y||_2=1}} ||x-y||_2.\hspace{3in} (*) \end{align}
(I have chosen the L2 norm for concreteness, but any norm will do). It is easy to see that $(*)$ will be non-zero.
In my particular case, $X$ is a linear subspace and $Y$ is a degree-two determinantal(-ish) variety. I have been working with Macaulay2, so a possible implementation with this software would be amazing.
Also, is there a way to phrase this question more elegantly, perhaps in terms of minimizing the distance between the projectivizations of $X,Y$ with respect to a projective metric? Please forgive my ignorance of projective metrics. I would appreciate an introductory reference in this vein.