Assume we have sets of the form

$$
M_j = \{x\in\mathbb{R}^d : f_j(x) \le 0,x \ge 0\}
$$

where $x\ge 0$ means $x_i \ge 0 \quad \forall i=1,\dots, d$. 

# Goal

I am looking for an (explicit) representation of a function $f$, such that the minkowski sum $M$ is of the same form as the $M_j$, i.e.

$$ 
M = \sum_{j=1}^n M_j = \{x\in\mathbb{R}^d : f(x) \le 0,x\ge 0\}
$$
where a Minkowski sum of sets is defined as

$$
   A+B := \{x+y : x\in A, y\in B\}.
$$

# Inuition Building

## Linear boundaries

Consider $f_j(x) = \langle w_j, x\rangle - b_j$. For $d=2$, we are really considering triangles.

| d=2 | d=3 |
| ---- | ---- |
| [![d=2 visualization][1]][1]| [![d=3 visualization][2]][2]

Now in the case $d=2$, $n=2$ with linear $f_j$ boundaries the resulting set should intuitively look like this:

[![d=2, n=2 Minkowski sum][3]][3]

This is because you can choose the maximum $x_2^{(2)}$, where
$$
    x_2^{(j)} := \max \{x_2\in\mathbb{R}^d : (0, x_2) \in M_j\},
$$
from the set $M_2$ and add $M_1$ to get everything left of $x_1^{(1)}$, and then add $(x_1^{(1)}, 0)$ from $M_1$ to the entire set of $M_2$ to get the rest. That this is not just a subset of $M$ is intuitively clear, but more difficult to formalize.

Similarly for $n=4$ we have

[![d=2, n=4 minkowski sum][4]][4]

So in some sense we are sorting the $w_j$ (or $f_j$) more generally.

## Continuous Concave Boundaries

Intuitively this should generalize for concave, continuous $f_j$. This is because we can approximate $f_j$ with a polygonal chain and we know that we can then find sets with linear borders such that their minkowski sum has this polygonal chain as the border
$$
   M_j \approx \sum_{l=1}^k M_{jk} = \sum_{l=1}^k \{ x\in \mathbb{R}^d : \langle w_{jk}, x\rangle - b_{jk} \le 0, x\ge 0\}
$$
So we get
$$
   \sum_{j=1}^n M_j \approx \sum_{j=1}^n \sum_{l=1}^k M_{jk}
$$
which reduces this to the linear case. Making the approximation better and better should result in an equality in the limit.

# Question

It feels like someone should have thought about this sort of thing already. But I can not find any sources. Probably because I don't know the names to look for. I looked a bit into linear programming, but since this is not an optimization problem per se I found it difficult to translate this into that. I am really looking for good notation to make this problem easier to think about.

At the moment I would struggle to translate the pictures into formal proofs - I might eventually manage to prove that my guess of $M$ is in fact $M$, but I do not see how that generalizes to higher dimensions where I can not just guess $M$ intuitively. So I am looking for a way where you can simply start calculating and $M$ is a result. That approach would likely generalize to higher dimensions.

# Application

The application I have in mind is justifying the [Production possibility fontier](https://en.wikipedia.org/wiki/Production%E2%80%93possibility_frontier) in economics, by combining the production capabilities of multiple individuals. This frontier is generally postulated to look like this:

https://en.wikipedia.org/wiki/Production%E2%80%93possibility_frontier#/media/File:Production_Possibilities_Frontier_Curve.svg

But no reason is provided (cf. https://economics.stackexchange.com/q/17501/15245). While our two dimensional case with four sets already has striking similarity, and if you assume a large populace with randomly distributed $w_j$ you can probably show that you get this type of curve in the limit.

  [1]: https://i.sstatic.net/z5mc4.jpg
  [2]: https://i.sstatic.net/uGKSJ.jpg
  [3]: https://i.sstatic.net/CnDxum.jpg
  [4]: https://i.sstatic.net/SqI0b.jpg