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 $\ell^2$ 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 $α\in (1, 2) \cup (2, \infty)$. This implies that on every nonzero inner product space, the $α$-power generalization of the Hornich-Hlawka inequality holds for all triples precisely when $α\in (0,1] \cup \{2\}$. For the remaining two problems, we establish reductions. We prove that $\ell^p(N)$ is Hornich-Hlawka for all $N \geq 3$ and $2 < p \leq \log 3 /(\log 3 - \log 2)$ if and only if it is Hornich-Hlawka at $N = 3$ and $p = \log 3 /(\log 3 - \log 2)$. Finally, for $p(\cdot): (0, 1) \to [1, 2]$ measurable and integers $k \geq 0$, we establish a Sobolev-Nakano equivalence: we prove that $W^{k,p(\cdot)}(0,1)$, equipped with the sum of the Luxemburg norms of its derivatives of orders $0, \ldots, k$, is Hornich-Hlawka if and only if $L^{p(\cdot)}(0,1)$ is Hornich-Hlawka.