The Minkowski problem based on the (p,q)-mixed quermassintegrals

TL;DR

This paper introduces the (p,q)-mixed quermassintegrals, a new unified concept in convex geometry, and explores the Minkowski problem related to these integrals, including properties and inequalities.

Contribution

It establishes the (p,q)-mixed quermassintegrals via (p,q)-dual mixed curvature measure and investigates the associated Minkowski problem.

Findings

Derived properties of (p,q)-mixed quermassintegrals

Established geometric inequalities for these integrals

Provided insights into the Minkowski problem for (p,q)-dual measures

Abstract

Lutwak, Yang and Zhang [23] introduced the concept of Lp dual curvature measure for convex bodies and star bodies, and studied the Minkowski problem. We in this paper establish a new unified concept, in briefly, the (p,q)-mixed quermassintegrals, via (p,q)-dual mixed curvature measure, and further have a deep discussion on Minkowski problem with respect to the (p,q)-dual mixed curvature measure. By the way, we derive at some important properties and geometric inequalities for (p,q)-mixed quermassintegrals.