Hornich–Hlawka failure for three-dimensional ℓp spaces

Prove that the three-dimensional sequence space ℓp(3) is not Hornich–Hlawka for every exponent 2 < p ≤ p0, where p0 = log 3/(log 3 − log 2).

Background

The paper studies which finite-dimensional ℓp spaces satisfy the Hornich–Hlawka inequality. The known results establish the property for 1 ≤ p ≤ 2 and show failure for p > p0, leaving the intermediate range 2 < p ≤ p0 unresolved. Because failure for ℓp(3) propagates to ℓp(N) for every N ≥ 3, resolving this conjecture would settle the corresponding region of the classification table negatively.