Let $A_{1}, A_{2}, \ldots, A_{n}$ be points in $\mathbb{R}^{3}$ and $\pi_{*}$ be the least squares plane, i. e.
$$ \sum \limits_{i = 1}^{n}\rho^{2}(A_{i}, \pi_{*}) = \min_{\pi}\sum \limits_{i = 1}^{n}\rho^{2}(A_{i}, \pi) $$
Is it true that
$$ \sum \limits_{i: \rho(A_{i}, \pi_{*}) > 0}|\rho(A_{i}, \pi_{*})| = \sum \limits_{i: \rho(A_{i}, \pi_{*}) < 0}|\rho(A_{i}, \pi_{*})| $$ ?
Or can any other similiar proposition be proved about the sets of points with indices $\{i: \rho(A_{i}, \pi_{*}) > 0\}$ and $\{i: \rho(A_{i}, \pi_{*}) < 0\}$?
Can you advise any literature about these sets of points? Did any investigation ever been performed?
For points $(1, 0), (-1, 0), (0, a)$ it is true. Evidently LS line is $y = y_{min}$, where $y_{min}$ minimizes the following functional:
$$ I(y) = y^{2} + y^{2} + (y - a)^{2} $$ with $y_{min} = \frac{1}{3} a$. Hence, first sum is $\frac{2}{3} a$ and second sum is $\frac{1}{3} a + \frac{1}{3} a = \frac{2}{3} a$.
But I have doubt that it is true in general...
Yes, it's true.
Plane $\pi_{*}$ is minimizing the following functional:
$$ I(a, b, c, d) = \sum \limits_{i = 1}^{n}\frac{(a x_{i} + b y_{i} + c z_{i} + d)^{2}}{a^{2} + b^{2} + c^{2}} \to inf $$
thus
$$ \frac{\partial I}{\partial d} = \sum \limits_{i = 1}^{n}\frac{2 (a x_{i} + b y_{i} + c z_{i} + d)}{a^{2} + b^{2} + c^{2}} = \frac{2}{\sqrt{a^{2} + b^{2} + c^{2}}}\sum \limits_{i = 1}^{n}\frac{2 (a x_{i} + b y_{i} + c z_{i} + d)}{\sqrt{a^{2} + b^{2} + c^{2}}} = \frac{2}{\sqrt{a^{2} + b^{2} + c^{2}}} \sum \limits_{i = 1}^{n} \rho(A_{i}, \pi) = 0 $$
that's why
$$ \sum \limits_{i = 1}^{n} \rho(A_{i}, \pi_{*}) = 0 $$
and
$$ \sum \limits_{i: \rho(A_{i}, \pi_{*}) > 0}|\rho(A_{i}, \pi_{*})| = \sum \limits_{i: \rho(A_{i}, \pi_{*}) < 0}|\rho(A_{i}, \pi_{*})| $$