The dual Minkowski problem for symmetric convex bodies

The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on an even prescribed measure on the unit sphere for it to be the $q$-th dual curvature measure of an origin-symmetric convex body in $\mathbb{R}^n$. A full solution to this is given when $1 < q < n$. The necessary and sufficient condition is an explicit measure concentration condition. A variational approach is used, where the functional is the sum of a dual quermassintegral and an entropy integral. The proof requires two crucial estimates. The first is an estimate of the entropy integral proved using a spherical partition. The second is a sharp estimate of the dual quermassintegrals for a carefully chosen barrier convex body.