An extension of polynomial integrability to dual quermassintegrals

TL;DR

This paper generalizes the concept of polynomial integrability from bodies with polynomial parallel section functions to the setting of dual quermassintegrals, providing new characterizations and addressing smoothness considerations.

Contribution

It extends the characterization of polynomially integrable bodies to dual quermassintegrals and explores related smoothness issues.

Findings

Generalized polynomial integrability to dual quermassintegrals

Provided new characterizations of such bodies

Addressed smoothness problems in the generalized setting

Abstract

A body $K$ is called polynomially integrable if its parallel section function $V_{n-1}(K\cap\{\xi^\perp+t\xi\})$ is a polynomial of $t$ (on its support) for every $\xi$. A complete characterization of such bodies was given recently. Here we obtain a generalization of these results in the setting of dual quermassintegrals. We also address the associated smoothness issues.