We define the holomorphically convex hull of a set $M \subset D \subset \mathbb{C}^n$ to be the set $$\widehat{M} : = \{ z \in D : \left| f(z) \right| \leq \| f \|_M \ \forall f \in \mathscr{O}(D) \},$$ where $\| f \|_M : = \sup_{z \in K} \left| f(z)\right|$, for all compact sets $K \subset D$ and $\mathscr{O}(D)$ denotes the space of holomorphic functions on the domain $D$.
I have absolutely no intuition for what this definition means.
I am comfortable with the notion of a convex hull, but not sure how this is a generalisation of that, if it is.
Any assistance is appreciated.
Consider the standard convex hull that you are familiar with. The convex hull of a set $K$ is the smallest convex set that contains $K$. We may rephrase this to mean the intersection of all half spaces that contain $K$ or even better, we may define this to be the set of all points such that all $(\mathbb{C}$-)linear maps $\ell$ which take values not exceeding their maximum on $K$. Formally, we may then write the convex hull as $$\text{conv}(K) = \{ z \in \mathbb{C}^n : \left| \ell(z) \right| \leq \| \ell \|_K\},$$ for all complex linear functionals on $K$.
Replacing $\ell$ by holomorphic functions, we obtain the definition for a holomorphically convex hull.
Exercise: Prove that the holomorphically convex hull is contained in the (linear-)convex hull.