In How does the Herglotz trick work?, is explained as in "Proofs from THE BOOK" by Aigner and Ziegler, but the "trick" itself I found to be not so clear.
The trick says:
It follows from $(4)$ that $h\left(\frac{x_0}{2}\right)+h\left(\frac{x_0+1}{2}\right)=2m$, and hence that $h\left(\frac{x_0}{2}\right)=m$.
But I can't understand why $h\left(\frac{x_0}{2}\right)=m$, that implies $h\left(\frac{x_0}{2}\right)=h\left(\frac{x_0+1}{2}\right)$, and I don't find anything that justifies this.
What am I missing?
Did $h$ have maximum value $m$? That would explain the implication I think.