Properties of the Complement of a Sumset

Question

Let $(G, +)$ be a finite abelian group. Let $A, B$ be subsets of $G$. Then we define the sumset (or Minkowski sum) as: $$A + B = \{a + b\mid a\in A, b\in B\}$$ Fix a subset $C\subseteq G$. For any $A\subseteq C$, we can (attempt to) define the complement of $A$ (denoted $\overline{A}$) within $C$ as follows: $$\overline{A} := \arg\max_{A'\subseteq C}(A + A' \subseteq C)$$ This name is meant to contrast with the traditional set complement $A^c$, which can be written (where $\sqcup$ is the disjoint union): $$A^c := \arg\max_{A'\subseteq C}(A \sqcup A' \subseteq C)$$ Has this notion been introduced before? If so, have properties of $\overline{A}$ been worked out? I've tried leafing through some textbooks on Additive combinatorics, but haven't seen anything similar (despite it being a rather naïve notion to introduce).