In $\mathbb{R}^2$, a circle refers to a set of the form $\{(x,y)\mid (x-a)^2+(y-b)^2=r^2\}$ and a disc refers to a set of the form $\{(x,y)\mid (x-a)^2+(y-b)^2\leq r^2\}$. I am thankful that the terminology is crystal clear here.
What about other terminologies such as a square, or more generally, a polygon? Does a square only include the boundary points? More precisely, which one of the following is defined to be a square:
- $\{(x,y)\mid -1\leq x\leq 1, -1\leq y\leq 1\}$, or
- $\left([-1,1]\times\{\pm 1\}\right)\bigcup\left(\{\pm 1\}\times[-1,1]\right)$?
What is the name of the other set of the above two?
Edit: I was teaching my daughter ($2$ years old) and am surprised that most of her books call a disc a "circle". Likely those books are not written by professional mathematicians.
You might consider adjectives here. E.g. solid square vs. empty square.
Convex polytopes, and thus convex polygons too, are most often assumed to be defined as the hull of their vertices, i.e. as being the intersection of half-spaces. Then those would be solid, for sure.
--- rk