$$v_{\alpha}(\beta_{\alpha}) = \phi v_{\alpha}(1). \label{eq:zdloc2}$$

The dimensionless functions $\{a_{\alpha}\}$ are generalizations of eq \[eq:area\] for each wedge $\alpha$ with wedge height $R_{\alpha}$. These functions encode the geometry of the particular structure, and the cross-sectional area at the top of each wedge, $A_{\alpha}(R_{\alpha})$, scales as $R_0^{d-1}$, for a $d$-dimensional structure ([*e.g.*]{} $d\!=\!1$ for lamellae, $d\!=\!2$ for cylinders, $d\!=\!3$ for spheres). Expressing eq \[eq:interf\],\[eq:fstr\] in terms of $\{a_{\alpha},
A_{\alpha}\}$ and $R_0$, we minimize over $R_0$ to find the following free energy per chain of a particular structure: $$f = f_0 \left\{S_a^2 \left(
{S_{\scriptscriptstyle A}\over \varepsilon\phi^2} + {\varepsilon
S_{\scriptscriptstyle B} \over (1-\phi)^2}
\right)\right\}^{1/3}, \label{eq:avg}$$ where $$\begin{aligned}
S_{a} &=& {{\displaystyle \sum_{\alpha} 
A_{\alpha}(R_{\alpha}) a_{\alpha}(\beta_{\alpha})}
\over {\displaystyle \sum_{\alpha} 
v_{\alpha}(1)} R_{\alpha} A_{\alpha}(R_{\alpha})}
\label{eq:Sa} \\
\noalign{\vskip7pt}
S_{\scriptscriptstyle A} &=& {{\displaystyle \sum_{\alpha} 
R_{\alpha}^3 A_{\alpha}(R_{\alpha})
I_{\scriptscriptstyle A}(\alpha,\beta_{\alpha})} \over
{\displaystyle \sum_{\alpha} 
v_{\alpha}(1)} R_{\alpha} A_{\alpha}(R_{\alpha})}, \label{eq:SA}\end{aligned}$$ where $I_{\scriptscriptstyle A}(\alpha,\beta_{\alpha})$ is obtained from eqs \[eq:IA\]-\[eq:IB\] by substituting $a_{\alpha}(\beta_{\alpha})$ in place of $a(\beta)$, and a similar relation defines $S_{\scriptscriptstyle B}$.

By specifying the volume fraction in each wedge according to eq \[eq:zdloc2\], we locally satisfy the constraint arising from the fixed composition of the copolymers. In contrast, Likhtman and Semenov [@LikhSeme94] satisfied this constraint only globally within a particular structure, which would be relevant 