Motivation
Suppose that $u \in \mathbb{R}^d$ is a unit-norm vector, $\|u\| = 1$, $a, b, c$ are some positive constants and $\xi \in [0,1]$ is another constant (usually chosen close to 1). I am interested in solving the following problem
$$ \sup_{v \in \mathbb{R}^d}\ (\xi + (1-\xi)\|v\|_1^2)\left(\sqrt{ \frac{a \langle u, v \rangle^2 + b}{\xi + (1-\xi)\|v\|_1^2}} - c \right) $$
subject to $\|v\| = 1$.
Question
While any suggestions on how to find an optimal $v$ are welcome, I am specifically interested in the following question.
How to find a vector $v$ that maximizes $\langle v, u \rangle$ and satisfies $\|v\|_2 = 1$ and $\|v\|_1 = x$?
A related question (that may be an easier one): given a vector $v_1$ that maximizes $\langle v_1, u \rangle$ and satisfies $\|v_1\|_2 = 1$ and $\|v_1\|_1 = x$ and another vector $v_2$ that maximizes $\langle v_2, u \rangle$ and satisfies $\|v_2\|_2 = 1$ and $\|v_2\|_1 = x + \delta$, how to find the change $\langle v_1, u \rangle - \langle v_2, u \rangle$ as a function of $\delta$?