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check the image here

check the image here

For any fixed $\mathbf v \in \mathbb S^{m-1}$, we have $$||\mathbf v^* \mathbf a| - | \mathbf v^* \mathbf a' || \le | \mathbf v^* (\mathbf a - \mathbf a') | \le \| \mathbf a - \mathbf a' \|_2. \tag{3.2.23}$$ So, the function $f(\mathbf a) = | \mathbf v^* \mathbf a |$ is $1$-Lipschitz. A quick calculation shows that for $\mathbf a \sim \operatorname{uni} (\mathbb S^{m-1})$, we have $$\mathbb E [ (\mathbf v^* \mathbf a)^2] = \frac 1m. \tag{3.2.24}$$

I am confused about the Equation 3.2.34, can somebody help me, thanks!

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All of this relies on the isotropy of the uniform distribution of a vector on the unit sphere. Align your coordinate axis with the unit vector $v$, so that $v\cdot a=a_1$. Then use that $\mathbb{E}[a_i^2]$ is independent of the index $i$, so that $m\operatorname{\mathbb{E}} [a_1^2] = \sum_{i=1}^m \mathbb{E}[a_i^2] = \mathbb{E}[\sum_{i=1}^m a_i^2]=1$, hence $\mathbb{E}[v\cdot a]=\mathbb{E}[a_1^2]=1/m.$

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