Let $C$ be a convex region in the plane with area 1 that contains distinct points $p_1,\dots,p_n$. Say I'd like to divide $C$ into $n$ pieces $C_1,\dots,C_n$, each of area $1/n$, and I'd like to minimize the average Euclidean distance between region $C_i$ and point $p_i$: $$\min_{C_1,\dots,C_n}\sum_{i=1}^n\iint_{C_i}\|x-p_i\|dx$$ It is easy to show that the boundaries at the optimal solution must be hyperbolic arcs, like this:

Now, let's say I'd like to solve the same problem, but minimizing the average *squared* Euclidean distance:
$$\min_{C_1,\dots,C_n}\sum_{i=1}^n\iint_{C_i}\|x-p_i\|^2dx$$
it is also easy to show that the optimal boundaries are straight lines and form a weighted Voronoi diagram:

My question is: how different, in general, are these two solutions? More formally, if I take the optimal solution to one problem and evaluate its cost under the other problem, is there a bound as to how big the difference can be? When I coded this it looks like the gap is not very big.