Sharpening the triangle inequality : envelopes between L2 and Lp spaces

Inequalities that sharpen the triangle inequality for sums of $N$ functions in $L^p$

“…which is valid for all 0≤γ ≤1, is equivalent to the inequality proved in [4, Theorem 1.3], as noted in [8]. Indeed, this inequality for all p≥2 is within a few percent of being an identity, and thus there is little numerical difference between (4.2) and (4.3).…”

“…This leaves open, however, the possibility of replacing (1+Γ 2/p p ) p−1 by some other function of Γ p . After this paper was completed we received the preprint [8] in which such an inequality was proved:…”

“…Indeed, this inequality for all p≥2 is within a few percent of being an identity, and thus there is little numerical difference between (4.2) and (4.3). The significant difference is that the function of Γ p on the right side of (4.3) is shown to be the best possible in [8] in that for all γ ∈[0, 1], on any "nice" measure space, there are functions f and g for which Γ p (f, g)=γ, and such that equality holds in (4.3). However, even knowing that (4.3) is a non-improvable result for N =2, one does not have a new proof of (4.2) that does not rely on [4,Theorem 1.3].…”

“…His proposal partially motivated this study. The paper [8] also contains a bound on the L p norm of a sum of N functions for N >2, but it is for the case p∈ (1,2), and is of an entirely different character than the one we present here.…”

Complex Hanner's Inequality for Many Functions

“…Hanner's inequality has been influential and its various generalisations and sharpenings have been extensively studied, see, e.g. [1,2,3,9,16].…”