Bellman VS Beurling: sharp estimates of uniform convexity for $L^p$ spaces

TL;DR

This paper uses the Bellman function method combined with differential geometry to derive sharp estimates for the moduli of convexity in Lebesgue spaces, specifically through classical Hanner inequalities.

Contribution

It introduces a novel approach to find the Bellman function without guesswork, providing sharp convexity estimates for $L^p$ spaces.

Findings

Derivation of classical Hanner inequalities using Bellman functions

Sharp estimates for the moduli of convexity in Lebesgue spaces

Application of differential geometry to simplify Bellman function construction

Abstract

We obtain the classical Hanner inequalities by the Bellman function method. These inequalities give sharp estimates for the moduli of convexity of Lebesgue spaces. Easy ideas from differential geometry help us to find the Bellman function using neither "magic guesses" nor calculations.