Algebraically integrable bodies and related properties of the Radon transform

TL;DR

This paper reviews the current understanding of algebraically integrable bodies and their connection to the Radon transform, addressing Arnold's longstanding problems in geometric analysis.

Contribution

It provides a comprehensive overview of the state of Arnold's problems on algebraically integrable domains and their relation to the Radon transform.

Findings

Summary of known results on algebraically integrable bodies

Connections between integrability and Radon transform properties

Open problems and future directions in the field

Abstract

Generalizing Lemma 28 from Newton's ``Principia", Arnold asked for a complete characterization of algebraically integrable domains. In this paper we describe the current state of Arnold's problems. We also consider closely related problems about the Radon transform of indicator functions.