Suppose vectors $a_p\in \mathbb{R}^n$, $p\in [n]$ form an orthonormal basis. For a given unit vector $b\in \mathbb{R}^n$, the summation $\sum_{p\in[n]} b^Ta_p$ is at its maximum ($=\sqrt{n}$) if $b^Ta_p = \frac{\sqrt{n}}{n}$ for all $p\in[n]$.
Given unit vectors $b, c\in \mathbb{R}^n$, my question is what is a tight upper bound for the following summation $s=\sum_{p\in[n]} b^Ta_p c^Ta_p$ assuming $b^T c = \alpha$.
(for simplicity, we can assume both unit vectors $b$ and $c$ are in the positive orthant.)
