There is a set of points. For simplicity, let's say that those are 2D points (although the problem works in more dimensions as well). The goal is to find the minimum possible length of a connected 1-dimensional curve, such that it passes through all the points.

Intuitively, the curve will consist of straight lines, but how do those lines look?

To make things clear, here is an approximation for a solution for a square:

.     .
 \   / 
  ---  
 /   \ 
.     .

For an equilateral triangle, we would pick a point inside and connect all vertices to it. We can find the best point with some differential equations.

The problem is not that hard to solve for these generic shapes, but is there an algorithm that solves it for any set of points?

I find this very similar to minimal surfaces, the only difference is that these aren't surfaces, but rather curves.

Consider a set of points. For simplicity, let's say that those are 2D points (although the problem works in higher dimensions as well). The goal is to find the minimum possible length of a connected 1-dimensional curve, such that it passes through all the points.

Intuitively, the curve will consist of straight lines but how do those lines look like?

To make things clear, here is an approximation for a solution for a square:

.     .
 \   / 
  ---  
 /   \ 
.     .

For an equilateral triangle, we would pick a point inside and connect all vertices to it. We can find the best point with some differential equations.

The problem is not that hard to solve for these generic shapes. Is there an algorithm that solves the problem for any set of points?

I find this very similar to minimal surfaces -- the only difference is that these aren't surfaces but rather curves.