I have a collection of points $P_1, \ldots, P_n$ in 3D space, and I want to fit a plane to these points. Clearly this is a minimization problem -- I want to minimize some "total measure" of the distance from the points to my plane.
Suppose $d_i$ is the distance from the point $P_i$ to my plane. I can think of (at least) three quantities that I could try to minimize: $$ \text{Minimize} \quad D_1 = \sum_{i=1}^n d_i \quad \text{ giving me a plane} \;\pi_1 $$ $$ \text{Minimize} \quad D_2 = \sum_{i=1}^n d_i^2 \quad \text{ giving me a plane} \;\pi_2$$ $$ \text{Minimize} \quad D_3 = \max_{1\le i \le n} d_i \quad \text{ giving me a plane} \;\pi_3$$ My question: Are there any relationships between $D_1, D_2, D_3$ or between their associated planes $\pi_1, \pi_2, \pi_3$?
The reason I ask is that $\pi_3$ is what I really care about, but $D_2$ looks like it would be much easier to minimize. I’m wondering whether finding $\pi_2$ will help me find $\pi_3$.