Relabeling a strict Hlawka failure so its total-sum size is largest

Let $\iota$ be a finite index set and, for a real exponent $p$, write $\|v\|_p := \big(\sum_{i\in\iota} |v_i|^p\big)^{1/p}$ for the coordinate power-sum functional of a real vector $v:\iota\to\mathbb R$, and call $\|v\|_p$ the $p$-size of $v$ (it is a norm for $p\ge1$; this argument does not use the triangle inequality). For real vectors $x,y,z:\iota\to\mathbb R$ and a real constant $K$, put

and call $(x,y,z)$ a strict Hlawka failure at level $K$ when

equivalently — since this quantity equals $K(2S-P)-(S-T)$ — when $(x,y,z)$ violates the inequality $S-T\le K(2S-P)$ that the construction seeks to establish.

Given $K\ge 1$ and a strict Hlawka failure $(x,y,z)$ at level $K$, this theorem produces a new triple $u,v,w:\iota\to\mathbb R$ that is again a strict Hlawka failure at level $K$, and for which the total-sum size dominates every individual size:

This is the relabeling step in normalizing a hypothetical counterexample to the diagonal Hlawka bound: among the four sizes $\|x\|_p,\|y\|_p,\|z\|_p,\|{-(x+y+z)}\|_p$ associated with the zero-sum quadruple $x,y,z,-(x+y+z)$, it arranges for the total-sum size $\|u+v+w\|_p$ to be at least each of the three retained singleton sizes, while preserving the fact that the relabeled triple is still a strict failure.

Formalization Note No hypothesis constrains the exponent $p$: the argument is a purely algebraic rearrangement of the four sizes and of the deficit quantity above, valid for every real $p$. Accordingly the $p$-size is an arbitrary power-sum functional here, not necessarily a norm.