Denote $\|\cdot \|_p$ for the norm in $\ell_p^n$, where $1 \leq p \leq \infty$, and $n \geq 1$. Let $(x^\star_i)$ denote a nonincreasing arrangement of the sequence $(|x_i|) \in \mathbb{R}^n$.
We define $$h_{p, n, \delta}(x) = \sup_{\|y\|_2 \leq \delta, \|y\|_p \leq 1} \langle x, y \rangle, $$ which is the support functional of the convex body $\delta B_2^n \cap B_p^n$.
Fix $p \in [1, 2]$ and $\delta > 0$. Define \begin{gather*} C(p, n, \delta) = \inf_{x \neq 0} \frac{h_{p, n, \delta}(x)}{\delta \|x_{\leq m}^\star\|_2 + \|x_{\geq m}^\star\|_q}, \quad \mbox{where} \quad \\[1.5ex] \frac{1}{q} + \frac{1}{p} = 1, \quad \frac{1}{\alpha} = \frac{1}{p} - \frac{1}{2}, \quad \mbox{and} \quad m = [\delta^{-\alpha}]. \end{gather*} Here, we denote $v_{\leq t}$ (resp., $v_{\geq t}$) for the first $t$ (resp., last $n - t + 1$) coordinates of $v \in \mathbb{R}^n$.
Question: By Hölder's inequality, $C(p, n, \delta) \leq 1$. On the other hand, [1, Thms 4.1 and 4.2] imply $C(p, n, \delta) > 0$, independently of $n$. Are more explicit and quantitative estimate possible? Specifically:
(i) What is a (ideally, simple/quantitative) proof of $\inf_{n \geq 1} C(p, n, \delta) > 0$?
(ii) Can $C(p, n, \delta)$ or $\inf_{n \geq 1} C(p, n, \delta)$ be computed explicitly? If not, can we have more explicit, multiplicative approximations of them?
References:
[1] Holmstedt, Tord. Interpolation of quasi-normed spaces. Math. Scand. 26 (1970), 177--199. MR0415352
