MathNet 2023-part-07 — 187 problems (EN–VI) Question 100

Let $A$ and $B$ be points in space for which $AB = 1$. Let $\mathcal{R}$ be the region of points $P$ for which $AP \leq 1$ and $BP \leq 1$. Compute the largest possible side length of a cube contained within $\mathcal{R}$.