# No, there are only small empty convex polygons

@BillyJoe has discovered that Balogh, Gonzalez-Aguilar and Salazar (1) solved this question in 2012. They showed that a random set of points contains, on average, an empty convex polygon of size

$$\mathbb E[f]=\Theta\left(\frac{\log n}{\log\log n}\right)$$
They also show that with high probability, this is the size of the largest one, that is, the outliers do not influence the average very much (outliers exist of course: you could lay out $n$ points along a circle, but such configurations are apparently very rare).

Let me give an exposition of their surprisingly simple argument that you are likely to find such a large empty convex polygon.

First: The probability that $r$ points form a convex polygon is known (exactly! due to Valtr [2]), if the points are drawn from any parallelogram, namely it is
$$ \mathbb P[r \text{ points are convex}]=\left(\frac{\binom{2r-2}{r-1}}{r!}\right)^2$$

They now imagine that we get to repeatedly draw $r$ points at random from a parallelogram; if we have an $r$-sided convex polygon, the experiment is a success; otherwise, we repeat the experiment on a clean slate.
Then they ask: How many times would we have to repeat this experiment, drawing $r$ points each time, before we saw a polygon with $r$ sides?
It turns out that choosing $n$ satisfying $r=\frac{\log(n)}{2\log\log(n)}$ does the trick: then we do $n/r$ independent experiments before we see our first convex $r$-sided polygon, drawing a grand total of $n$ points over all experiments. So we have our bound.

Very good, but this would only work if we had $n/r$ disjoint parallelograms to work with.
But we don't, because in our setting we draw all our points from the $[0,1]\times [0,1]$ square, so our $n/r$ "experiments" are not independent. How do we solve this?

The authors use a clever trick here.
Namely, suppose that we have drawn $n$ points at random, and assume wlog that no two of them lie on a vertical line.
Then we may group the points from left to right in groups of $r$ points, so that the square is cut up into $n/r$ long rectangles, as in the illustration below.

These points lie in non-overlapping rectangles.
Moreover, conditioned upon the fact that a given rectangle contains $r$ points, they have been drawn uniformly at random from that rectangle, and so our bound applies anyway!

Q.E.D.

They then prove that this is in fact *the largest* polygon you are likely to find, but via a more involved argument, which I leave for the reader to explore.

Let me remark that my conjecture of $\mathbb E [f]=\Theta(\sqrt n)$ was *exponentially* higher than the true answer; a big error.
I had guessed that the largest empty convex polygon might contain roughly as many corners as the convex hull of the $n$ points, which I thought would be $\approx \sqrt n$ points, since that is the length of each side. This was the right direction, except that this estimate was *also* way off, since apparently most of the points near the edge are not in the convex hull after all, indeed only $\Theta(\log n)$ points comprise the hull, proved by Renyi and Sulanke [3]. Again, I was off by an exponential magnitude.

It seems to me that the estimate of the number of vertices that one likely encounters, $\geq \log(n)/2\log\log n$, can probably be improved (by a constant factor only, since the bound is tight asymptotically), by considering a more complex argument which tries to squeeze more points into each rectangle.
Currently, the upper and lower bounds of Balogh et al.'s differ by a factor of 320, namely they showed that w.h.p., with $r$ the largest empty convex polygon,
$$\frac{\log n}{2\log \log n}\leq r\leq 160\frac{\log n}{\log\log n}$$
It also seems that no attempts have been made to reduce this gap, so any budding Ramsey theory enthusiasts have their work cut out for them.

**References**

(1) *Balogh, József; González-Aguilar, Hernán; Salazar, Gelasio*, **Large convex holes in random point sets**, Comput. Geom. 46, No. 6, 725-733 (2013). ZBL1271.52003.

(2) *Valtr, P*, **Probability that (n) random points are in convex position**, Discrete Comput. Geom. 13, No. 3-4, 637-643 (1995). ZBL0820.60007.

(3) *Rényi, Alfréd; Sulanke, R.*, **Über die konvexe Hülle von (n) zufällig gewählten Punkten**, Z. Wahrscheinlichkeitstheor. Verw. Geb. 2, 75-84 (1963). ZBL0118.13701.