Chord Measures in Integral Geometry and Their Minkowski Problems

TL;DR

This paper introduces a new family of geometric measures called chord measures, explores their Minkowski problems, and provides solutions for these problems, including cases without regularity assumptions.

Contribution

It adds a novel family of measures to integral geometry and addresses their Minkowski problems, including fully nonlinear PDEs and solutions without regularity.

Findings

Introduction of chord measures as a new geometric measure family

Formulation of Minkowski problems for these measures and their variants

Solutions to major cases of these Minkowski problems without regularity assumptions

Abstract

To the families of geometric measures of convex bodies (the area measures of Aleksandrov-Fenchel-Jessen, the curvature measures of Federer, and the recently discovered dual curvature measures) a new family is added. The new family of geometric measures, called chord measures, arises from the study of integral geometric invariants of convex bodies. The Minkowski problems for the new measures and their logarithmic variants are proposed and attacked. When the given data is sufficiently regular, these problems are a new type of fully nonlinear partial differential equations involving dual quermassintegrals of functions. Major cases of these Minkowski problems are solved without regularity assumptions.