Hlawka's inequality in $\ell_p$ for $2\le p\le\log 3/\log(3/2)$

For a real exponent $p\ge1$ and $v\in\mathbb R^n$ write $\|v\|_p=\big(\sum_i|v_i|^p\big)^{1/p}$ (lpNorm p). Hlawka's inequality (also called the Hornich–Hlawka inequality) for $\|\cdot\|_p$ asserts that for all $x,y,z\in\mathbb R^n$,

In the platform's notation this is HasHlawkaConstant (lpNorm p) 1: the triple gap $\|x\|+\|y\|+\|z\|-\|x+y+z\|$ is at most the sum of the three pair gaps $\|x\|+\|y\|-\|x+y\|$. The threshold exponent is

the exponent at which $3^{1/p}=3/2$.

Statement (open). For every real $p$ with $2\le p\le p_W$ and every finite dimension $n$, Hlawka's inequality holds in $\ell_p(n)$ over $\mathbb R$.

Together with the failure for $p>p_W$, this would make $p_W$ the exact threshold for Hlawka's inequality in $\ell_p$. Marinescu and Niculescu ask in their Problem 1 to prove that $\ell_p(3)$ is not Hornich–Hlawka for any $p\in(2,\infty]$. This statement asserts the opposite on $(2,p_W]$.

Numerical evidence: multi-start maximisation of the ratio of triple gap to pair-gap sum over $\mathbb R^3$, with 400 to 1500 starts per exponent, finds a maximum of exactly $1$ for $2<p\le 2.7$ and a strict violation just above $p_W$. By the platform theorem real_bound_of_fin_three, the case $n=3$ already implies every $n$.