Holomorphically convex domain

Remarks

Since $\hat K_D$ is closed in $D$, the condition says that $\hat K_D$ has no limit points on the boundary of $D$.

Example: a convex open set $D$. For compact $K \subseteq D$, the hull $\hat K_D$ lies in the convex hull $\operatorname{ch}(K)$, which is compact and contained in $D$. A subset of $D$ that is closed in $D$ and contained in the compact set $\operatorname{ch}(K) \subseteq D$ is compact.

Non-example: $D = \set{z \in \C^2 : 0 < \abs{z} < 1}$ and $K = \set{\abs{z} = 1/2}$. By Hartogs's extension theorem every holomorphic $f$ on $D$ extends to the ball, and by the maximum principle $\abs{f(z)} \le \max_K \abs{f}$ for $0 < \abs{z} \le 1/2$. So $\hat K_D$ contains $\set{0 < \abs{z} \le 1/2}$, which is not compact.

By the Cartan–Thullen theorem, a domain in $\C^n$ is holomorphically convex if and only if it is a domain of holomorphy.