AI 中文总结
该研究基于顶点表示的多面体构造凸体空间的有限维逼近，推导定量估计并构造Galerkin序列，最终建立带约束全局优化问题有限维逼近的收敛性。
AI 中文摘要
我们引入基于顶点表示的多面体对凸体空间的有限维逼近。对于给定的一组方向，可容许的点构型构成欧几里得向量空间中的一个多面凸子锥，由有限组线性不等式描述。该锥的内部仅包含非退化表示，其中所有参数点均为所表示多面体的不同顶点。我们研究该参数锥的几何性质、其定义不等式中的冗余性，以及凸体到所得多面体空间的自然投影。我们根据给定方向覆盖单位球的密集程度推导定量逼近估计，并构造嵌套的Galerkin序列，其逼近误差局部一致收敛至零。最后，我们利用这些逼近性质建立凸体空间中带约束全局优化问题的有限维逼近的收敛性。
英文摘要
We introduce finite-dimensional approximations to the space of convex bodies based on polytopes in vertex representation. For a prescribed set of directions, the admissible point configurations form a polyhedral convex subcone of the Euclidean vector space, described by a finite system of linear inequalities. The interior of this cone contains only nondegenerate representations, in which all parameter points are distinct vertices of the represented polytope. We study the geometry of the parameter cone, redundancies in its defining inequalities, and natural projections of convex bodies onto the resulting polytope spaces. We derive quantitative approximation estimates in terms of how densely the prescribed directions cover the unit sphere and construct nested Galerkin sequences whose approximation error converges locally uniformly to zero. Finally, we use these approximation properties to establish the convergence of finite-dimensional approximations of constrained global optimization problems in the space of convex bodies.
