What's the name of a closure of a complement of a closed halfspace?

Question

Let $A$ be a closed half-space of an affine space. What's the name of $cl(A')$? I was tentatively using the name "conjugate", but I don't know if its correct.

Answer 1

You could call it the "complementary half-space" (as used, e.g., here or here). It might be worth specifically calling out that you're taking the closed complementary half-space, or specifically writing down that you're turning $a^{\mathsf T}x \le b$ into $a^{\mathsf T}x \ge b$, just to be clear - but it's not unusual terminology.

The word "conjugate" has a specific meaning in convex analysis, so if you were to use it, I'd assume you were talking about something like the dual cone.