Core Characterized by Absorbing Translations

Let $X$ be a vector space and $\Omega\subset X$ be nonempty. For $x\in X$, consider the translate $\Omega-x:=\{w-x\mid w\in\Omega\}$ (using set difference/translation notation).

Proposition: A point $x\in X$ lies in core(Ω) if and only if $\Omega-x$ is an absorbing set, i.e., for every $v\in X$ there exists $\lambda>0$ such that $v\in \alpha(\Omega-x)$ for all scalars $\alpha$ with $|\alpha|\ge \lambda$.

Context: This characterization turns the "two-sided line" definition of the core into a scaling condition that is often easier to check in practice.