I need to understand the proof of a theorem from a book called "geometric group theory by Graham and Roller". The theorem says:
Let $p:[0,1]\rightarrow \mathbb{R}^n$ be $C^1$-path. Then there exists an open subset $A\subset[0,1]$ such that $\int_A p'(t)dt=\frac{1}{2}\int_0^1 p'(t)dt(=\frac{1}{2}p(1))$.
and the proof is:
Every point $s=(t_1,...,t_n)\in \mathbb R^n$ lying on the (topological) sphere $\sum^n_{i=1} |t_i|=1$ defines a partition of $[0,1]$ into two subsets $[0,1]=A_+^s \cup A_-^s$, as $[0,1]$ is covered by $n$ intervals and by definition $t\in A_+^s $ if it is contained in such interval, say $[|t_1|+...+|t_i|,|t_1|+...t_{i+1}|]$ where $t_{i+1}\geq0$ and $t \in A_-^s$ if $t_{i+1}$ corresponding to $t$ is negative. Then we have a (continuous!) map of our sphere to $\mathbb R^n$ given by $s\mapsto \int_{A_+^s} p'(t)dt -\int_{A_-^s} p'(t)dt$ which is by Borsuk-Ulam theorem vanishes at some $s$ in the sphere. Then we take the smallest two sets $A_+^{s_0}$ and $A_-^{s_0}$ for $A$.
If you pick some $n$ numbers $t_i$, with $\sum_i |t_i|=1$, then you get an increasing sequence $q_i = \sum_{j=0}^{i}|t_i|$, $$ 0 = q_0 \leq q_1 \leq q_2 \leq \cdots \leq q_n = 1. $$ So the $q_i$ partition the interval $[0,1]$ into $n$ intervals. Then you take the intervals $[q_{i-1}, q_i]$ and put them into either $A_-$ or $A_+$ depending on the sign of $t_i$.
The point of doing this is that the map $$f:(t_1,\ldots,t_n)\mapsto \int_{A_+}p'(t)\,dt - \int_{A_-}p'(t)\,dt $$ is a continuous map from the ball $\{(t_i)\}$ to $\mathbb{R}^n$, so you can apply the Borsuk-Ulam theorem to it, and use the fact that $f(-t_i) = -f(t_i)$. Once you do, all you have left is to note that $A_+ = [0,1] \setminus A_-$, because of how $A_\pm$ are defined.
Does this answer your question?