Let $P$ and $Q$ be two polygons in $mathbb{R}^2$. Given $a > 0$, denote by $aP$ its image under the dilation by $a$ centered around the origin (i.e. the polygon obtained by replacing each vertex $(p_0,p_1)$ by the vertex $(ap_0, ap_1)$). Let $C(P,Q)$ denote the convex hull of the union $P cup Q$, and let $N(P,Q)$ denote the number of new edges created (i.e. edges in $C(P, Q)$ which do not appear in $P$ or $Q$). Let $P+Q$ denote the Minkowski sum of the two polygons.

Based on some empirical results, I was led to the below heuristic (if $X$ is a Gaussian centered at $0$, then $|X|$ follows the half-normal distribution).

**Heuristic:** Let $a,b$ be chosen from the half-normal distribution, and $P, Q$ be two fixed convex polygons in $mathbb{R}^2$ that are “sufficiently generic”. Then:

$$ mathbb{E}(N(aP, bQ))<10$$

Unfortunately it turns out that this does not hold for all choices of polygons $P$ and $Q$, and finding a counterexample is not difficult (see below for details). Above $10$ is an arbitrarily chosen constant, the key point is that it does not depend on the number of vertices in the polygons $P$ and $Q$. Let a Gaussian random polygon $P$ be the convex hull of $n$ random, independent points in $mathbb{R}^2$ sampled according to the standard normal distribution, for some $n$; here $n$ is the “size” of the polygon. So this leads me to my real question:

**Question:** What are some conditions which ensure that the above inequality holds? These conditions should be “generic” in some sense, and should be satisfied (with high probability) when $P$ and $Q$ are Gaussian random polygon of fixed size.

In particular, an answer to the above question should rigorously prove that the heuristic holds (with high probability) when $P$ and $Q$ are Gaussian random polygons. The next question is a more general version, with Minkowski sums.

**Question 2:** Let $a_1, cdots, a_m, b_1, cdots, b_n$ be chosen from a half-normal distribution, and $P_1, cdots, P_m, Q_1, cdots, Q_n$ be polygons in $mathbb{R}^2$. Consider the following statement:

$$ mathbb{E}(N(a_1P_1+cdots+a_mP_m, b_1Q_1+cdots+b_nQ_n)) < 10$$

What are some conditions on these polygons which ensure that the above inequality holds? These conditions should be “generic” in some sense, and should be satisfied (with high probability) when the $P_i$ and the $Q_j$ are Gaussian random polygon of fixed sizes.

If the phrasing of the above questions are too vague, I’d be happy with a proof that the above inequalities hold, with high probability, when the polygons are random Gaussian polygons. One special case I’m interested in is when the sizes of the Gaussian polygons is 2 (i.e. they are line segments).

The counterexample for Q1 can be constructed as follows. Suppose $O P_1 P_2 cdots P_n$ is a convex polygon, with the vertices in clockwise order. Let $P_i P_{i+2}$ intersect $O P_{i+1}$ at the point $Q_{i+1}$. The points should be chosen carefully so that the ratio $frac{OQ_{i+1}}{OP_{i+1}} < epsilon$, where $epsilon>0$ is small (this can be done inductively). Let $P$ be the polygon $OP_2 P_4 …$ and $Q$ the polygon $OP_1 P_3 …$. With this choice, it is easy to see that $mathbb{E}(N(aP, bQ))$ grows linearly with $n$.