Polygon-point Shortest distance

Dear Dan, I have a quest for you! :D

An algorithm that finds the point on an arbitrary polygon that is the shortest from an arbitrary point.
You could use that for a lot of things, like simulating a mace or a rope attached to an object in 2d or 3d etc.
And then you could advance this to a crazy level, like making it work for infinite dimensions.
But you would need to understand what a polygon is in 4d and above.
If you do this, you are the king of abstract linear algebra and geometry.
Prerequisites: Linear algebra -> Projection , Geometry, 2-vector-norm, subspaces, Abstract math genius skills, And a lot of aspirin