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

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 ${\mathrm{\ell }}^2$ with no three-dimensional Hornich-Hlawka subspace. Second, we present a Nakano space that is not Hornich-Hlawka. Third, we show that the $\alpha$-power generalization of the Hornich-Hlawka inequality fails to hold on nonzero normed spaces when $\alpha \in (1,2)\cup (2,\mathrm{\infty })$. This implies that on every nonzero inner product space, the $\alpha$-power generalization of the Hornich-Hlawka inequality holds for all triples precisely when $\alpha \in (0,1]\cup \{2\}$. For the remaining two problems, we establish reductions. We prove that ${\mathrm{\ell }}^p(N)$ is Hornich-Hlawka for all $N\ge 3$ and $2<p\le \log 3\mathrm{/}(\log 3-\log 2)$ if and only if it is Hornich-Hlawka at $N=3$ and $p=\log 3\mathrm{/}(\log 3-\log 2)$. Finally, for $p(\cdot ):(0,1)\to [1,2]$ measurable and integers $k\ge 0$, 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 $0,\dots ,k$, is Hornich-Hlawka if and only if $L^{p(\cdot )}(0,1)$ is Hornich-Hlawka.