Explicit solutions to Christoffel-Minkowski problems and Hessian equations under rotational symmetries

TL;DR

This paper provides explicit solutions to the Christoffel-Minkowski problem and related Hessian equations for convex bodies with rotational symmetry, using measure conditions on spherical caps and balls.

Contribution

It introduces explicit formulas for solutions under rotational symmetry and addresses existence problems for mixed area measures and Hessian equations.

Findings

Explicit support functions for convex bodies of revolution are derived.

Solutions are constructed for mixed Monge-Ampère equations with radial symmetry.

The Dirichlet problem for k-Hessian equations is explicitly solved in this context.

Abstract

An explicit solution to the Christoffel-Minkowski problem for convex bodies of revolution is presented. The conditions on the prescribed measure involve only first moments over spherical caps, and the support function of the resulting convex body is given by an explicit representation formula in terms of the measure. More generally, existence problems for mixed area measures are addressed. The approach relies on constructing explicit convex solutions to mixed Monge-Amp\`ere equations on $\mathbb{R}^n$ under the assumption of radial symmetry, with the conditions on the measure being expressed through its values on open balls. As a special case, the Dirichlet problem for $k$-Hessian equations on $\mathbb{R}^n$ is treated.