Dual Curvature Density Equation with Group Symmetry
Károly J. Böröczky, Ágnes Kovács, Stephanie Mui, Gaoyong Zhang
TL;DR
This work extends the $L_p$ dual Minkowski problem to convex bodies with group symmetry, formulating a Monge-Ampère type equation on the sphere and solving it via a variational method for $q>0$ and $-q^*<p<0$. By developing a framework for $G$-invariant convex bodies and establishing sharp dual-quermassintegral estimates, the authors obtain existence results beyond origin-symmetric cases. The key contribution is a direct variational proof that, under suitable symmetry and integrability assumptions on the data, yields a $G$-invariant convex body $K$ whose $L_p$ dual curvature measure matches a given $G$-invariant measure. This broadens the scope of solvable instances of the continuous $L_p$ dual Minkowski problem and underscores the role of group symmetry in convex geometric PDEs.
Abstract
This paper studies the general Lp dual curvature density equation under a group symmetry assumption. This geometric partial differential equation arises from the general Lp dual Minkowski problem of prescribing the Lp dual curvature measure of convex bodies. It is a Monge-Ampere type equation on the unit sphere. If the density function of the dual curvature measure is invariant under a closed subgroup of the orthogonal group, the geometric partial differential equation is solved in this paper for certain range of negative p using a variational method. This work generalizes recent results on the Lp dual Minkowski problem of origin-symmetric convex bodies.