How to find an embedded submanifold in $\mathbb{R}^{n}$ which pass through some given points in $\mathbb{R}^{n}$ and has the lowest dimension?

Question

How to find an embedded submanifold in $\mathbb{R}^{n}$ which pass through some given points in $\mathbb{R}^{n}$ and has the lowest dimension?

I have no idea with this question. I know, at least, the manifold like this must exist because $\mathbb{R}^{n}$ itself is an embedded manifold which pass through these points, there is a manifold that has the lowest dimension and pass through these points.

Answer 1

If is a finite number of points you can construct a curve simply by interpolation.