Marinescu–Niculescu Problem 1 (real exponents): no $\ell_p(3)$ with $p>2$ is Hornich–Hlawka

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 (as posed). Problem 1 of Marinescu–Niculescu asks to prove that none of the spaces $\ell_p(3)$ with $p\in(2,\infty]$ is Hornich–Hlawka. This entry formalises the finite real exponents $p\in(2,\infty)$: for every real $p>2$, Hlawka's inequality fails in $\ell_p(3)$. The case $p=\infty$ is already settled by the triple above.

The survey establishes the range $p>p_W$. The range $2<p\le p_W$ is what the problem leaves open. A disproof needs a single exponent in $(2,p_W]$ at which $\ell_p(3)$ is Hornich–Hlawka, for example via hlawka_holds_five_halves.