Many results about mass partitions are proved by lifting $\mathds {R}^d$ to a higher-dimensional space and dividing the higher-dimensional space into pieces. We extend such methods to use lifting arguments to polyhedral surfaces. Among other results, we prove the existence of equipartitions of $d+1$ measures in $\mathds {R}^d$ by parallel hyperplanes and of $d+2$ measures in $\mathds {R}^d$ by concentric spheres. For measures whose supports are sufficiently well separated, we prove results where one can cut a fixed (possibly different) fraction of each measure either by parallel hyperplanes, concentric spheres, convex polyhedral surfaces of few facets, or convex polytopes with few vertices.

中文翻译：

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质量分配问题中的提升方法

通过将$\mathds {R}^d$ 提升到高维空间并将高维空间分割成碎片，证明了许多关于质量分区的结果。我们将此类方法扩展到使用提升参数到多面体表面。在其他结果中，我们通过平行超平面证明了 $\mathds {R}^d$ 中 $d+1$ 度量的均分的存在，以及通过同心圆证明了 $\mathds {R}^d$ 中 $d+2$ 度量的均分。领域。对于支持充分分离的度量，我们证明了可以通过平行超平面、同心球体、少数面的凸多面体表面或具有少数顶点的凸多面体来切割每个度量的固定（可能不同）部分的结果。