Let $u, v \in \mathbb{S}^{d-1}$ be two unit vectors with $u \cdot v \geq c_1$. Let $Z \in \{-1, +1\}^d$ be a random sign vector where each coordinate is +1 or -1 independently with probability 1/2.

I would like to compute a lower bound, if one exists, on the conditional expectation $$\mathbb{E}[\langle Z, v\rangle|\langle Z, u\rangle \geq c_2]$$ in terms of $c_1$ and $c_2$. It seems like some kind of coupling argument would work here since the conditioning makes $Z$ more likely to be close to $v$ overall, but I haven't been able to work out the details.

**Edit**: The specific regime I am studying is that of $c_1$ and $c_2$ both being relatively small but positive, and am looking for a positive lower bound (given that the unconditional expectation is zero).

**Edit**: If $u$ is a normalized sign vector, I've been able to solve the problem as follows: define $Z' = Z \odot u$ (elementwise product), and let $v' = v \odot u$. By symmetry, $Z'$ is also a random Rademacher vector.

Then, our condition translates to $\sum_i v'_i \geq c_1$, and the expectation can be written as $\mathbb{E}[\langle Z', v'\rangle|\sum_i Z_i' \geq c_2]$. For every integer $k > c_2$, the expectation $\mathbb{E}[\langle Z', v'\rangle|\sum_i Z_i' = k]$ is just a rescaling of $\sum_i v'_i$, which is positive by assumption.

However, I haven't been able to extend this beyond $u$ being a sign vector.