  Consider two boundary subregions $A$ and $B$ that lie in a common boundary
Cauchy surface, and consider also the associated HRT surface $\gamma_B$ for
$B$. In that context, the constrained HRT surface $\gamma_{A:B}$ can be defined
as the codimension-2 bulk surface anchored to $A$ that is obtained by a maximin
construction restricted to Cauchy slices containing $\gamma_B$. As a result,
$\gamma_{A:B}$ is the union of two pieces, $\gamma^B_{A:B}$ and $\gamma^{\bar
B}_{A:B}$ lying respectively in the entanglement wedges of $B$ and its
complement $\bar B$. Unlike the area $\mathcal{A}\left(\gamma_A\right)$ of the
HRT surface $\gamma_A$, at least in the semiclassical limit, the area
$\mathcal{A}\left(\gamma_{A:B}\right)$ of $\gamma_{A:B}$ commutes with the area
$\mathcal{A}\left(\gamma_B\right)$ of $\gamma_B$. To study the entropic
interpretation of $\mathcal{A}\left(\gamma_{A:B}\right)$, we analyze the
R\'enyi entropies of subregion $A$ in a fixed-area state of subregion $B$. We
use the gravitational path integral to show that the $n\approx1$ R\'enyi
entropies are then computed by minimizing $\mathcal{A}\left(\gamma_A\right)$
over spacetimes defined by a boost angle conjugate to
$\mathcal{A}\left(\gamma_B\right)$. In the case where the pieces
$\gamma^B_{A:B}$ and $\gamma^{\bar B}_{A:B}$ intersect at a constant boost
angle, a geometric argument shows that the $n\approx1$ R\'enyi entropy is then
given by $\frac{\mathcal{A}(\gamma_{A:B})}{4G}$. We discuss how the $n\approx1$
R\'enyi entropy differs from the von Neumann entropy due to a lack of
commutativity of the $n\to1$ and $G\to0$ limits. We also discuss how the
behaviour changes as a function of the width of the fixed-area state. Our
results are relevant to some of the issues associated with attempts to use
standard random tensor networks to describe time dependent geometries.
