On the maximal hyperplane in [formula omitted]

Abstract

In this paper we introduce the family of projections Pf,α:ℓpn→ker⁡fPf,α=Id−1∑i=1n|fi|αf⊗〈f〉α−1 and study their maximal p-norms. Note that α=2 and α=q corresponds to orthogonal and polar projections respectively. For fixed α>1, we are able to compute maxf⁡||Pf,α||p and show that, for α≠q, any maximizer f has at most two non-zero coordinates different. For α=q there are more maximizers but we can always choose one that has at most two non-zero coordinates different. Based on that we compute orthogonal λorthH(ℓpn) and polar λpolH(ℓpn) hyperplane constants of ℓpn. Furthermore, the general formula for maxf⁡||Pf,α||p allows us to show that for n=3 and any hyperplane H=ker⁡fλ(H=ker⁡f,ℓpn)≤λ(ker⁡1,ℓpn), i.e., ker⁡1 is a maximal hyperplane in ℓp3, confirming the n=3 case of the long standing conjecture.