# Does optimal fitting flat must pass through the mean of the point set?

I am confused about a statement made in the paper Linear Time Algorithm for Projective Clustering, section 5.1, second paragraph, second line.

Project clustering is a natural generalization of k-means clustering. Given a point set $$P$$, the goal to find $$k$$, $$j$$-dimensional affine spaces(flats) $$F_1,\cdots,F_k$$, such that the following objective function is minimized

$$\sum_{p\in P} \min_{1\leq i \leq k}dist^2(p,F_i)$$, where $$dist(p,F)$$ is the orthogonal distance between point $$p$$ and flat $$F$$.

Clearly, as is the case with k-means, the $$k$$ optimal flats induces a partitioning of the point set $$P$$, into $$k$$ subsets $$C_1,\cdots,C_k$$, such that points in $$C_i$$ is closer to flat $$F_i$$ than any other flat $$F_j, i\neq j$$.

The aforementioned paper (in page 7, third line from last), it is said that "It is easy to see that $$F_i$$ passes through the mean $$o_i$$ of $$C_i$$". I fail to see the simplicity of this. Can someone please elaborate on that, with a proof sketch?

N.B: mean of a point set $$P$$ is defined as $$\frac{1}{|P|}\sum_{p\in P}p$$.

• See the book Foundations of Data Science cs.cornell.edu/jeh/book.pdf, section 3.9.1. – Chandra Chekuri Apr 15 '19 at 20:59
• @ChandraChekuri ok, I get it. In fact I have managed to prove something similar. I will post this as an answer tomorrow. – Sudipta Roy Apr 15 '19 at 21:26