Universal (ℓp, ℓq)-flattenability for q > 2 and p < q

Determine whether for every pair of parameters with 1 ≤ p < q and q > 2, every finite graph G is (ℓp, ℓq)-flattenable; that is, whether all edge-length vectors induced by realizations in ℓq can also be realized in ℓp for this parameter regime.

Background

Using known embeddings between Lp spaces, the paper proves that for 1 ≤ p ≤ q ≤ 2, every graph is (ℓp, ℓq)-flattenable. This relies on isometric embeddings of ℓq into Lq[0,1] and of Lq[0,1] into Lp[0,1] for q ≤ 2.

For q > 2, the same embedding approach fails (e.g., ℓq3 does not embed in ℓ1 for q > 2), leaving open whether universal flattenability from ℓp to ℓq persists beyond the range q ≤ 2.

References

It is unknown to the authors if every graph is $(\ell_p, \ell_q)$-flattenable when $q > 2$ and $p < q$.

— Edge-length preserving embeddings of graphs between normed spaces (2405.02189 - Dewar et al., 3 May 2024) in Section 4.2 (Varying p)