Disk in $\mathbb R^2$ with uniform norm I am having a trouble understanding a definition.
The points are in $\mathbb R^2$, and the author defines $\delta(p, r)$ to be an $l_\infty$ disk of radius $r$ centred at $p$. I just learned what the $l_\infty$ norm is, from Wikipedia. However I don't understand what the $\delta$ notation means.
Can someone help me visualize how this disk would look like? If possible, by elaborating the properties of the $l_\infty$ norm.
Thanks!
 A: $\delta(p,r)$ is a square with sides of length $2r$ and centre at $p$. For simplicity I'm going to elaborate taking $p$ to be the origin but this carries through for any $p$ by translation (which is an isometry).
$$\delta(\vec{O},r) = \{ x \in \mathbb{R^2} : \| x \|_{\infty} < r \} = \{x \in \mathbb{R^2} : |\mbox{max}(x_i)| < r \}$$ where $x_i$s are co-ordinates of $x$. So the boundary of this disc is the set of points with at least one co-ordinate being $r$ in modulus but then this is a square with side length $2r$.
A: $\delta$ is just a symbol used to denote this type of set; the author could have used $B(p,r)$, $D(p,r)$, or really any symbol, as long as the notation is consistent. Don't let it bother you too much.
For a geometric understanding, the unit ball centered at $0$, $\delta(0,1)$, is the square of side length $2$ centered at $0$. If you write out the definition of $\delta(0,1)$ and think about it carefully, you should be able to deduce this. (A picture of this is shown in the Wikipedia article you linked.)
For contrast, you may also be interested in the $l_1$ distance and the balls associated to that norm. See https://en.wikipedia.org/wiki/Taxicab_geometry for an illustration of what the $l_1$ unit ball looks like.
