Dear Uncle Colin,

I was working on a MAT question that asked about finding a subset, $S$, of the 2D-plane, and a point $P$ such that no point in $S$ was the closest to $P$. I had no idea where to start!

Omitted Point, Euclidean Norm

Hello, OPEN, and thanks for your message!

My first answer wasn’t the one the MAT answer suggested, but I stand by it: if $S$ is the empty set, it’s definitely a subset of the 2D-plane ((every point in the empty set is also a member of the set of points in the real plane, vacuously)). And since there are no points in $S$ to be nearest to $P$, this satisfies the condition.

But they had something a bit trickier in mind.

According to the answer sheet, a valid answer would be something like:

$S$ is the set of points such that $x^2+y^2 < 1$, and $P$ is a distance greater than 1 from the origin.

Let’s unpack that a bit. The set $S$ is an open disc, a filled circle centred on the origin, with a radius of 1, but such that the circumference is not part of the set. $P$ is outside of the circle. So why is there not a point in $S$ that’s closest to $P$?

Suppose (without loss of generality) that $P$ is on the positive $x$-axis. Any candidate closest point would also have to be on the positive $x$-axis, with an $x$-coordinate smaller than 1 to be a member of $S$.

If someone claims “The point $(X,0)$, is the closest member of $S$ to $P$”, you can immediately say “No, it isn’t – the point $\left(\frac{1+X}{2},0\right)$, midway between $P$ and the circumference, is inside of $S$ and closer to $P$ than your point.” Contradiction. Boom.

It’s a bit subtle (which is why I prefer my empty set method), and I’m not sure it’s a fair thing to ask an A-level student to come up with on the fly in an exam, but it’s interesting enough to write up.

I hope that helps!

- Uncle Colin