Nonnegativity of the Hlawka deficit's Hessian on the cyclic coordinate box

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). Let $\mathrm{pairTriple}(X)_j$ denote the sum of the two columns of $X$ other than $j$ (so $\mathrm{pairTriple}(X)_0=X_1+X_2$, $\mathrm{pairTriple}(X)_1=X_0+X_2$, $\mathrm{pairTriple}(X)_2=X_0+X_1$), and $\mathrm{totalTriple}(X)=X_0+X_1+X_2$. Let $\mathrm{cyclicCenter}$ be the triple whose $j$-th column has $-1$ in position $j$ and $1$ elsewhere, and $\mathrm{entryBox}:=\{X:|X_{j,i}-\mathrm{cyclicCenter}_{j,i}|\le19/100\text{ for all }j,i\}$.

For $v,h\in\mathbb R^3$, define

For $p,K\in\mathbb R$, define the Hessian of the Hlawka deficit at a triple $X$, in direction $Z$ (another triple), as

For every $p\ge256$, every $K$ with $1\le K\le p$, every $X\in\mathrm{entryBox}$, and every triple $Z$,

$\mathrm{deficitHessian}$ is built from $\mathrm{normHessian}$ with exactly the combinatorial pattern — three singleton terms with coefficient $2K-1$, one "total" term with coefficient $1$, and three "pair" terms with coefficient $-K$ — that a Hlawka-type deficit $(2K-1)\bigl(N(X_0)+N(X_1)+N(X_2)\bigr)+N(X_0+X_1+X_2)-K\bigl(N(X_0+X_1)+N(X_0+X_2)+N(X_1+X_2)\bigr)$ has in a size functional $N$. Differentiating each of these seven norm-terms twice along the line $X+sZ$ — using that $\mathrm{normHessian}_p$ is exactly the second derivative of the finite coordinate $p$-norm along a line, away from the origin — reproduces $\mathrm{deficitHessian}_{p,K}(X,Z)$ term by term. So this theorem is a statement about curvature: for every admissible $K$ in the stated range, the Hlawka-type deficit built from the finite coordinate $p$-norm has nonnegative second derivative, in every direction $Z$, at every point of the cyclic coordinate box.