I found an unclear part in derivation of PCA in the lecture notes of A. Bandeira for MIT Fall 2015 18.S096: Topics in Mathematics of Data Science course.
In $\S 1.1.1$ the author derives PCA as best $d$-dimensional affine fit as follows. We take data points $x_1,\ldots,x_n \in \mathbb{R}^p$ and search for their representation (as coordinates $\beta_k$) in a $d$-dimensional affine subspace defined by shift $\mu$ and orthogonal basis $V=[v_1,\ldots,v_d]$ via a least-squares fit problem: $$ \min\limits_{\mu,\beta_k,V:V^TV=I}\sum\limits_{k=1}^{n}\|x_k - (\mu + V\beta_k)\|^2_2. $$
First we try to optimize for $\mu$ and use first-order condition $$\nabla_\mu\sum\limits_{k=1}^{n}\|x_k - (\mu + V\beta_k)\|^2_2=0,$$ for which to hold we need that $$\left(\sum\limits_{k=1}^{n}x_k\right) -\mu n - V\left(\sum\limits_{k=1}^{n}\beta_k\right)=0.$$
Question:
At this point the author says that $\sum\limits_{k=1}^{n}\beta_k=0$ and goes on with the proof (which is fine), however I could find no rigorous reason why this should be true. Simple examples suggest such a fact (say, take two 2d points and fit a line to them), but I would appreciate an explanation.