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Let ${[X_0,X_1]}_t$, 0 ≤ t ≤ 1, be Banach spaces obtained via complex interpolation. With suitable hypotheses, linear operators T that act boundedly on both $X_0$ and $X_1$ will act boundedly on each ${[X_0,X_1]}_t$. Let $T_t$ denote such an operator when considered on ${[X_0,X_1]}_t$, and $\sigma \left(T_t\right)$ denote its spectrum. We are motivated by the question of whether or not the map $t\to \sigma \left(T_t\right)$ is continuous on (0,1); this question remains open. In this paper, we study continuity of two related maps: $t\to {\left(\sigma \left(T_t\right)\right)}^{\wedge }$ (polynomially convex hull) and $t\to {\partial }_e\left(\sigma \left(T_t\right)\right)$ (boundary of the polynomially convex...