On analytic characterization of convex sets in $\mathbb{R}^m$ (a survey)

TL;DR

This survey reviews analytic methods for characterizing convex sets in Euclidean spaces, covering both planar cases and higher dimensions, summarizing key theoretical results and their implications.

Contribution

It provides a comprehensive overview of analytic characterizations of convexity across different dimensions, consolidating existing results in a unified survey.

Findings

Analytic criteria for convexity in the plane

Extensions of these criteria to higher dimensions

Summary of key theoretical results in convex analysis

Abstract

In the first part of this note, we review results concerning analytic characterization of convexity for planar sets. The second part is devoted to results valid for arbitrary $m \ge 2$.