How to show after small change the intersection between two convex shapes still belongs to an open set

Question

Assume two convex shapes $A$ and $B$ in $\mathbb{R}^3$ (both $A$ and $B$ are compact sets in $\mathbb{R}^3$), and denote their intersection as $C = A \cap B$. Let $C$ belong to an open set $D$, i.e., $C \subset D$. How can we show that after small changes in the position and orientation to $A$ and $B$, which is denoted as $\bar{A}$ and $bar{B}$, respectively, their new intersection $\bar{C} = \bar{A} \cap \bar{B}$ still belongs to $D$, i.e., $\bar{C} \subset D$?