Suppose that $P$ is a Minkowski summand of $Q$. Define the barycentric subdivision $\operatorname{bsd}(Q;P)$ of $Q$ with respect to $P$ as follows. Start with $F=Q$, and proceed by decreasing dimension of faces $F\preceq Q$. Pull the translate $\varepsilon\kappa(F)-\varepsilon\widehat{\kappa(F)}+\widehat{F}$ of the corresponding face of $\varepsilon P$ in the relative interior of $F$.