The cyclic candidate $K_p$ is a necessary lower bound in dimension at least three

Let $n\ge3$ be a natural number and $p>1$ a real exponent. Consider the coordinate $\ell^p$-norm on $n$-tuples of complex numbers, $\|x\|_p = \bigl(\sum_{i=1}^n |x_i|^p\bigr)^{1/p}$ for $x\in\mathbb C^n$. For $x,y,z\in\mathbb C^n$, call

the pair deficit, $\mathrm{tripleGap}(x,y,z)=\|x\|_p+\|y\|_p+\|z\|_p-\|x+y+z\|_p$ the triple deficit, and $\mathrm{pairGapSum}(x,y,z)$ the sum of the three pair deficits over the pairs $\{x,y\},\{x,z\},\{y,z\}$. Call a real number $C$ an admissible Hlawka constant for $\|\cdot\|_p$ on $\mathbb C^n$ when

holds for all $x,y,z\in\mathbb C^n$. Let

be the cyclic ratio (for $t\ge0$, $A_p(t)$ is the coordinate $p$-norm of each of $(-t,1,1),(1,-t,1),(1,1,-t)\in\mathbb R^3$ and $B_p(t)$ that of each of their pairwise sums), and let

be the cyclic candidate constant.

The theorem states: if $C$ is any admissible Hlawka constant for $\|\cdot\|_p$ on $\mathbb C^n$ with $n\ge3$, then

This is the necessity half of sharpness for the diagonal construction. A separate theorem shows $K_p$ is itself admissible for complex diagonal triples in every finite dimension, for every real $p\ge256$. Combining the two: for $p\ge256$, $K_p$ is the sharp dimension-independent Hlawka constant for $\|\cdot\|_p$ on $\mathbb C^n$, $n\ge3$; this theorem alone, valid for every real $p>1$, only shows that no constant smaller than $K_p$ can work, and does not by itself establish admissibility or exact sharpness outside the range where the companion theorem is proved.

Formalization Note. This theorem is stated for the coordinate norm $\|\cdot\|_p$ on $\mathbb C^n$ directly. A separate theorem (schattenPNorm_diagonal) identifies $\|\cdot\|_p$ with the Schatten $p$-norm of a diagonal operator on $\mathbb C^n$; that identification is not needed to state or prove this theorem.