Balanced Core: Explicit Expression? Denote the collection of all balanced subsets by: $\mathcal{B}:=\\{B\subseteq X: B\text{ balanced}\\}$ Since the union of arbitrary balanced sets is balanced we can form the balanced core of arbitrary sets: $$\bigcup_{A\supseteq B\in\mathcal{B}}B$$ Is there a way to express the balanced core explicitely without refering to balanced sets?

Yes, first of all $0\in A$ iff $A^\star\
eq\varnothing$. Now for $0\in A$ it holds: $$\bigcup_{A\supseteq B\in\mathcal{B}}B=\bigcap_{|\kappa|\geq 1}\kappa A$$ (Note that $0\in A$ is absolutely necessary since $\cap_{|\kappa|\geq 1}\kappa (0,\infty)=(0,\infty)$ but $\cup_{(0,\infty)\supseteq B\in\mathcal{B}}B=\varnothing$.)

_See discussion on balanced core, page 80 in Horvath'sTopological Vector Spaces and Distribution._