Convexity in real analysis

TL;DR

This paper explores the concept of convexity in real analysis, providing definitions, characterizations, and applications that connect analytic and geometric perspectives.

Contribution

It offers a comprehensive analytic framework for convexity, linking it to classical geometric ideas and expanding its applications in analysis and geometry.

Findings

Definitions of convexity via functions and quadratic forms

Characterizations of various convexity notions

Applications to analysis and geometry

Abstract

We treat the classical notion of convexity in the context of hard real analysis. Definitions of the concept are given in terms of defining functions and quadratic forms, and characterizations are provided of different concrete notions of convexity. This analytic notion of convexity is related to more classical geometric ideas. Applications are given both to analysis and geometry.