An Extension of Polynomial Integrability to Dual Quermassintegrals

Abstract

Abstract A bounded domain $K$ 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 domains was given recently. Here we obtain a generalization of these results in the setting of dual quermassintegrals. We also address the associated smoothness issues.