Projecting unit $\ell_p$ vector onto $\ell_2$ and measuring distortion in $\ell_p$

Question

Let $p \ge 1$ and let $x \in \mathbb{R}^d$ be a unit vector in the $\ell_p$ norm: $\sum_{i=1}^d |x_i|^p = 1$. Let $v \in \mathbb{R}^d$ be a unit vector in $\ell_2$ in the same dimension: $\sum_{i=1}^d v_i^2 = 1$. Is there a tight upper bound for $$\max_{x, v} \, \|\langle v, x \rangle \cdot v \|_p ?$$

Answer 1

Yes, there is. As in the comments by Ryszard Szwarc above, the result is the supremum of a continuous function over a compact ball (in finite dimensions). By the Weierstrass extreme value theorem, the maximum is achieved at some point. If you want to calculate it explicitly, you can introduce Lagrange multiplier and find stationary points of the Lagrangian.