Draw geometric shapes dividing n-dimensional space in sectors closer to selected points

Question

Let's say I am on a 2-dimensional space, and I have two points $(x_0, y_0)$ and $(x_1, y_1)$. To divide the plane into two parts, the set of points closer to $(x_0, y_0)$ and the set of points closer to $(x_1, y_1)$ (with the usual metric), I simply draw the line that is equidistant from both and perpendicular to the segment connecting the two points. If my space is n-dimensional, I choose the (n-1)-hyperplane perpendicular to the segment. But, what if I have m points (m possibly different from n), how do I draw those subspaces? Even some steps with just 3 or 4 points to give me the idea would help.

Answer 1

The points in the plane that are closer to $P$ than to any of $Q,R,S$ constitute the intersection of 3 sets: the set of points closer to $P$ than to $Q$, the set of points closer to $P$ than to $R$, and the set of points closer to $P$ than to $S$. You already know how to produce each of those 3 sets, so just take their intersection. The result will be a polygon (in the plane) or polyhedron (in higher dimensions) bounded by (some of) the perpendicular bisectors that you already know about.

For lots more information, look up "Voronoi diagram" or "Voronoi tesselation".