Sharp complex coordinate Hlawka constant for p ≥ 89

For every real number $p \ge 89$ (any real, not necessarily an integer), the real number $K_p$ defined below is the least element of the set of real constants $C$ that satisfy the following condition. For every $n \in \{0,1,2,\dots\}$ and all $x, y, z \in \mathbb{C}^n$ (which need not be distinct or nonzero),

Addition is coordinatewise. The norm is $\|v\|_p := \big(\sum_{i=1}^{n}|v_i|^p\big)^{1/p}$, where $|v_i|$ is the complex modulus. Every power here is a real power of a nonnegative base, so for $p \ge 89$ this is the usual $\ell^p$ norm on $\mathbb{C}^n$. The bundle calls the left side the triple gap and the bracket the pair-gap sum. $C$ ranges over all real numbers, positive or negative.

"Least element" means two claims:

1. $K_p$ is admissible: the inequality holds with $C = K_p$ for every $n$ and all $x,y,z\in\mathbb{C}^n$.
2. No smaller uniform constant exists: if a real $C$ makes the inequality hold for every $n$ and all $x,y,z\in\mathbb{C}^n$, then $K_p \le C$.

The constant. For $t \in [\tfrac12, 2]$ set

and let $K_p := \sup\{R_p(t) : \tfrac12 \le t \le 2\}$. This is a supremum in $\mathbb{R}$ over the closed real interval.

Edge cases.

• $n = 0$: it is included but constrains nothing. The only vector is $0$, the empty sum gives $\|0\|_p = 0$, and the inequality reads $0 \le 0$.
• Scope of claim 2: it compares $K_p$ only with constants that work in every dimension at once. The statement does not say $K_p$ is the best constant for any single fixed $n$.
• Junk values: in the formal system, division by zero gives $0$, and the supremum of a set that is empty or unbounded above is $0$. Neither case arises here. On $[\tfrac12,2]$ we have $|1-t|\le 1$, so $(2t)^p + 2^p > 2 \ge 2|1-t|^p$, which is equivalent to $6A_p(t) - 3B_p(t) > 0$. So $R_p$ is continuous on the compact interval, and $K_p$ is its maximum value there.
• Range of $p$: nothing is claimed for $p < 89$.