# Simple linear interpolation in $\mathbb{R}^k$?

When doing simple linear interpolation on a data set in $\mathbb{R}^2$, I just sort the points by one co-ordinate and then do piecewise interpolation between a point and its successor in the sorted data set.

1. How would I do this in $\mathbb{R}^3$, and generally in $\mathbb{R}^k$?

2. How would I know which points to "connect"? (In the two-dimensional case the sorting takes care of that.)

3. How complex would it be to calculate such simple interpolations?

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