标题： 旋转对称下的Christoffel-Minkowski问题和Hessian方程的显式解 显示英文标题

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

摘要： 提出了旋转凸体的Christoffel-Minkowski问题的显式解。 规定的测度的条件仅涉及球形帽上的第一矩，所得到的凸体的支持函数通过测度的显式表示公式给出。 更一般地，处理了混合面积测度的存在性问题。 该方法依赖于在$\mathbb{R}^n$上构造混合Monge-Ampère方程的显式凸解，假设径向对称性，测度的条件通过其在开球上的值来表达。 作为特殊情况，处理了$\mathbb{R}^n$上的$k$-Hessian方程的Dirichlet问题。 显示英文摘要

摘要： 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.