Affine dual Minkowski problems

Abstract

While affine functionals of convex bodies and their affine isoperimetric inequalities have been extensively studied, the construction of geometric measures arising from affine geometric invariants (other than volume) has been missing.

In this work, affine ‘‘invariant’’ measures derived from the dual affine quermassintegrals are presented. Minkowski problems for the new affine-invariant measures are proposed and studied. The new variation formula derived here leads to new affine operators that map star bodies to star bodies. An affine isoperimetric inequality is obtained for new bi-dual intersection bodies.