Normalizing a strict Hlawka failure to unit norm-sum

Let $\iota$ be a finite index set and, for a real exponent $p>0$, write $\|v\|_p:=\big(\sum_{i\in\iota}|v_i|^p\big)^{1/p}$ for a real vector $v:\iota\to\mathbb R$. For $x,y,z:\iota\to\mathbb R$ and $K\in\mathbb R$, set $S=\|x\|_p+\|y\|_p+\|z\|_p$, $T=\|x+y+z\|_p$, $P=\|x+y\|_p+\|x+z\|_p+\|y+z\|_p$, and call $(x,y,z)$ a strict Hlawka failure at level $K$ when $(2K-1)S+T-KP<0$, equivalently $S-T>K(2S-P)$.

Given $p>0$, $K\ge1$, and a strict Hlawka failure $(x,y,z)$ at level $K$ — with no further hypothesis on $(x,y,z)$ — this theorem produces a new triple $u,v,w:\iota\to\mathbb R$ that is again a strict Hlawka failure at level $K$, for which the singleton norms sum to exactly one, and for which the total-sum norm dominates each singleton norm:

This produces, from an arbitrary strict counterexample, a canonically scaled one — normalized to unit singleton-norm sum and with total-norm domination — that fixes the scale used throughout the later scalar and coordinate estimates confining a hypothetical counterexample to the sharp diagonal Hlawka inequality.

Formalization Note For $0<p<1$ the triangle inequality can fail; the argument uses only positive-definiteness and positive homogeneity, which hold for every $p>0$.