Localizing a Hlawka failure to the cyclic coordinate box

For $p>0$ and $x:\{0,1,2\}\to\mathbb R$, write $\mathrm{lpNorm}_p(x):=\bigl(\sum_i|x_i|^p\bigr)^{1/p}$ for the finite coordinate $p$-norm. For $K\in\mathbb R$ and $x,y,z:\{0,1,2\}\to\mathbb R$, let

Writing $\mathrm{pairGap}(N)(a,b):=N(a)+N(b)-N(a+b)$, $\mathrm{tripleGap}(N)(a,b,c):=N(a)+N(b)+N(c)-N(a+b+c)$, and $\mathrm{pairGapSum}(N)(a,b,c)$ for the sum of the three pair gaps, one has $\mathrm{hlawkaDeficit}_{p,K}(x,y,z)=K\cdot\mathrm{pairGapSum}(\mathrm{lpNorm}_p)(x,y,z)-\mathrm{tripleGap}(\mathrm{lpNorm}_p)(x,y,z)$, which is $\ge0$ for all $x,y,z$ exactly when $\mathrm{lpNorm}_p$ has Hlawka constant $K$; so a negative value of $\mathrm{hlawkaDeficit}_{p,K}$ is a strict failure of that inequality at $(x,y,z)$.

Write a triple $X$ as three columns $X_0,X_1,X_2\in\mathbb R^3$, with $X_{j,i}$ coordinate $i$ of column $j$ (Lean: X j i), and let $\mathrm{tripleDeficit}_{p,K}(X):=\mathrm{hlawkaDeficit}_{p,K}(X_0,X_1,X_2)$. Let $\mathrm{cyclicCenter}$ be the triple whose $j$-th column has $-1$ in position $j$ and $1$ elsewhere, and let $\mathrm{entryBox}:=\{X:|X_{j,i}-\mathrm{cyclicCenter}_{j,i}|\le19/100\text{ for all }j,i\}$.

Let

For every $p\ge256$ and every $x,y,z:\{0,1,2\}\to\mathbb R$ with $\mathrm{hlawkaDeficit}_{p,K_p}(x,y,z)<0$, this theorem produces a triple $X\in\mathrm{entryBox}$ that is still a strict failure:

This localizes an arbitrary strict counterexample — after normalizing its total mass, relabeling which vector plays which coordinate role together with a sign flip per coordinate, and rescaling by $3$ — into the fixed box of triples within $19/100$ of the cyclic sign pattern. Turning an otherwise unbounded search for failures of the Hlawka inequality into one confined to a small, fixed region is what makes the region's own geometry (its convexity, and the curvature of the deficit on it) usable against the failure.

Formalization Note The box radius $19/100$ is the exact entrywise tolerance this construction's coordinate-geometry and curvature estimates are built around: the localization step used here puts the confined counterexample's entries within $6/50+84/(5p)$ of the cyclic pattern, which is $0.185625$ at $p=256$ — below $19/100=0.19$, but with little room to spare.