Constructions and reductions on the Hornich-Hlawka inequality and its generalizations

Abstract

We investigate five open problems concerning the Hornich-Hlawka inequality and its generalizations posed by Marinescu and Niculescu. We give constructive answers to three of them. First, we construct an equivalent norm on ℓ² with no three-dimensional Hornich-Hlawka subspace. Second, we present a Nakano space that is not Hornich-Hlawka. Third, we show that the α-power generalization of the Hornich-Hlawka inequality fails to hold on nonzero normed spaces when α∈ (1, 2) ∪ (2, ∞). This implies that on every nonzero inner product space, the α-power generalization of the Hornich-Hlawka inequality holds for all triples precisely when α∈ (0,1] ∪ {2}. For the remaining two problems, we establish reductions. We prove that ℓ^p(N) is Hornich-Hlawka for all N ≥ 3 and 2 < p ≤ log 3 /(log 3 − log 2) if and only if it is Hornich-Hlawka at N = 3 and p = log 3 /(log 3 − log 2). Finally, for p(·): (0, 1) → [1, 2] measurable and integers k ≥ 0, we establish a Sobolev-Nakano equivalence: we prove that W^k,p(·)(0,1), equipped with the sum of the Luxemburg norms of its derivatives of orders 0, …, k, is Hornich-Hlawka if and only if L^p(·)(0,1) is Hornich-Hlawka.

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BibTeX RIS

BibTeX

@misc{lee2026constructions,
  title = {{Constructions and reductions on the Hornich-Hlawka inequality and its generalizations}},
  author = {Seokbin Lee},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23785},
  eprint = {2610.23785},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23785},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Constructions and reductions on the Hornich-Hlawka inequality and its generalizations
AU  - Seokbin Lee
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23785
UR  - https://arxiv.org/abs/2610.23785
AB  - We investigate five open problems concerning the Hornich-Hlawka inequality and its generalizations posed by Marinescu and Niculescu. We give constructive answers to three of them. First, we construct an equivalent norm on ℓ^2 with no three-dimensional Hornich-Hlawka subspace. Second, we present a Nakano space that is not Hornich-Hlawka. Third, we show that the α-power generalization of the Hornich-Hlawka inequality fails to hold on nonzero normed spaces when α∈ (1, 2) ∪ (2, ∞). This implies that on every nonzero inner product space, the α-power generalization of the Hornich-Hlawka inequality holds for all triples precisely when α∈ (0,1] ∪ {2}. For the remaining two problems, we establish reductions. We prove that ℓ^p(N) is Hornich-Hlawka for all N ≥ 3 and 2 < p ≤ log 3 /(log 3 − log 2) if and only if it is Hornich-Hlawka at N = 3 and p = log 3 /(log 3 − log 2). Finally, for p(·): (0, 1) → [1, 2] measurable and integers k ≥ 0, we establish a Sobolev-Nakano equivalence: we prove that W^k,p(·)(0,1), equipped with the sum of the Luxemburg norms of its derivatives of orders 0, …, k, is Hornich-Hlawka if and only if L^p(·)(0,1) is Hornich-Hlawka.
ER  -

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