The Barrier Cone of a Convex Set and the Closure of the Cover

For an arbitrary non-empty closed convex set A in

, we prove that the polar of the difference between the barrier cone

\mathbb{B}(A) B ( A )

and its interior

\operatorname{int}(\mathbb{B}(A)) int ⁡ ( B ( A ) )

coincides with the recession cone

0^+(\operatorname{cl}(\mathbb{G}(A))) 0 + ( cl ⁡ ( G ( A ) ) )

of the closure of the cover

\mathbb{G}(a) G ( a )

.