Chord measures in integral geometry and their Minkowski problems

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.