# Sharpening the triangle inequality: envelopes between $L^{2}$ and $L^{p}$ spaces

## TL;DR

This paper refines and extends inequalities related to the triangle inequality in various $L^{p}$ spaces, providing optimal bounds and characterizations for sums of functions across a broad range of p-values.

## Contribution

It offers new upper and lower bounds for $
orm{f+g}_p^p$ in different $L^{p}$ spaces, generalizes to multiple functions, and characterizes equality cases, improving prior results.

## Key findings

- Established optimal bounds for $
orm{f+g}_p^p$ across all real p (excluding zero).
- Extended bounds to sums of multiple functions for $p \\in [1,2]$.
- Characterized cases of equality in the inequalities.

## Abstract

Motivated by the inequality $\|f+g\|_{2}^{2} \leq \|f\|_{2}^{2}+2\|fg\|_{1}+\|g\|^{2}_{2}$, Carbery (2006) raised the question what is the "right" analogue of this estimate in $L^{p}$ for $p \neq 2$. Carlen, Frank, Ivanisvili and Lieb (2018) recently obtained an $L^{p}$ version of this inequality by providing upper bounds for $\|f+g\|_{p}^{p}$ in terms of the quantities $\|f\|_{p}^{p}, \|g\|_{p}^{p}$ and $\|fg\|_{p/2}^{p/2}$ when $p \in(0,1] \cup [2,\infty)$, and lower bounds when $p \in (-\infty,0) \cup (1,2)$, thereby proving (and improving) the suggested possible inequalities of Carbery. We continue investigation in this direction by refining the estimates of Carlen, Frank, Ivanisvili and Lieb. We obtain upper bounds for $\|f + g\|_p^p$ also when $p \in (-\infty,0) \cup (1,2)$ and lower bounds when $p \in (0,1] \cup [2,\infty)$. For $p \in [1,2]$ we extend our upper bounds to any finite number of functions. In addition, we show that all our upper and lower bounds of $\|f+g\|_{p}^{p}$ for $p \in \mathbb{R}$, $p\neq 0$, are the best possible in terms of the quantities $\|f\|_{p}^{p}, \|g\|_{p}^{p}$ and $\|fg\|_{p/2}^{p/2}$, and we characterize the equality cases.