= Solution
The <Wulff construction> minimizes total orientation-dependent surface <free energy> at fixed crystal <volume>:
$$
G_{\rm surf}=\sum_i\gamma_iA_i,\qquad h_i=\Lambda\gamma_i.
$$
Planes at distances proportional to their <surface energies> bound the equilibrium shape. The construction is thermodynamic; observed growth rates need not select the same shape.
For a mixed-metal particle, minimize a more complete <free energy> including bulk mixing/ordering, Au–Ag interface <energy>, exposed-surface composition, adsorbates and elastic <strain>, while conserving the amount of each metal. Important parameters include orientation-dependent <surface energies>, metal chemical potentials, interface coherency, <lattice mismatch>, <temperature> and ligand coverage. <surface segregation> can change the exposed composition and hence the relevant $\gamma_i$.
\b[A generalized free-energy minimization can address a multicomponent particle; the elementary <Wulff construction> alone does not predict its core-shell composition.] Sequential Au-core formation and Ag deposition can preserve a kinetically trapped interface when diffusion is too slow to reach equilibrium. If shape relaxation is fast but interdiffusion slow, an external shape may approach a constrained minimum even though the complete particle is not at global equilibrium. Growth kinetics, ligand stabilization and finite-size effects limit a bare-surface equilibrium model. A faceted image is therefore insufficient proof that either shape or composition is an equilibrium state.
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