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The radius of a set $S \in \mathbb{R}^L$ is defined as: $$\operatorname{rad}(S) = \inf_{x \in \mathbb{R}^L} \sup_{y \in S} \|x-y\|$$

I interpret the definition as follows: the radius of $S$ is related to the smallest element in $\mathbb{R}^L$ (not necessarily in $S$) that leads to the highest (Euclidean) distance to an element $y \in S$.

Could someone please provide a (geometrical) intuition for this definition?

emacs drives me nuts
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user12632521
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1 Answers1

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Geometrical intuition is: pick an element $x\in\mathbb R^L$ and let $r_x$ be the radius of the smallest (closed) circle centered in $x$ that contains the whole $S$. One can see is that $||x-y||\le r_x$ for all $y\in S$, so $r_x$ is an upper bound of all the values $||x-y||$. As we wanted it to be the smallest such bound, we have:

$$r_x=\sup_{y\in S}||x-y||$$

So far so good - for every $x$ we've found the smallest (closed) circle centered in $x$ containing $S$. Now, let's move $x$ across $\mathbb R^L$, to find a point $x$ at which this radius is the smallest. And, in fact, we are not looking for the exact $x$ here - we are just looking for that smallest radius. Thus, we take:

$$r=\inf_{x\in\mathbb R^L}r_x=\inf_{x\in\mathbb R^L}\sup_{y\in S}||x-y||$$

and call that number the "radius" of $S$.

If, for example, $S$ is itself a circle with radius $\rho$, and $x$ is at distance $d_x$ from the centre, to cover the whole $S$ with the circle centered in $x$, you need the radius of the covering circle to be $r_x=d_x+\rho$ to reach the furthermost point of $S$. Obviously, across all points $x$ this will be the smallest if $x$ is the centre of $S$ and $d_x=0$, which means that the radius of $S$ will be $0+\rho = \rho$. This means that, for circles, the radius in this new sense corresponds to the radius in the usual sense.

You can check (as an exercise) that:

  • The radius of a "ring" between two circles with radii $r_1$ and $r_2$ ($r_1<r_2$) is $r_2$.
  • The radius of a square with side $a$ is $\frac{a\sqrt{2}}{2}$.
  • The radius of a line in a plane, or a plane in a 3D space etc. is infinite.
  • The radius of a single point is $0$.
  • The radius of an oval with semi-axes $a>b$ is $a$