Cutting the Same Fraction of Several Measures

2012/03/31 by Arseniy Akopyan, Роман Карасев, Roman Karasev · 1 citation
Chemistry · Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Chemistry #Combinatorics #Computational Geometry and Mesh Generation #Computer science #Data mining #Discrete mathematics #Fraction (chemistry) #Geometry #Hyperplane #Limits and Structures in Graph Theory #Mathematics #Measure (data warehouse) #Partition (number theory) #Regular polygon #Statistics #Value (mathematics) #math.CO #math.MG

paper · pdf · doi:10.1007/s00454-012-9450-4

published as Discrete and Computational Geometry 49:2 (2013), 402-410 · 7 pages 2 figures

arxiv created 2012/05/04 · openalex publication_date 2012/08/08 · arxiv updated 2013/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We study some measure partition problems: Cut the same positive fraction of d+1 measures in \mathbb R^d with a hyperplane or find a convex subset of \mathbb R^d on which d+1 given measures have the same prescribed value. For both problems positive answers are given under some additional assumptions.

Citations

• Using the Borsuk–Ulam Theorem
• Drei Sätze über die n-dimensionale euklidische Sphäre
• Convex Equipartitions of volume and surface area
• Equipartition of several measures
• BALANCED CONVEX PARTITIONS OF MEASURES IN ℝ d
• Generalized “sandwich” theorems
• Theorems of Borsuk-Ulam type for flats and common transversals of families of convex compact sets

Cited by

• Splitting Sandwiches Unevenly via Unique Sink Orientations and Rainbow Arrangements

Related

• Measure Partitions Using Hyperplanes with Fixed Directions
• Cookie cutters: Bisections with fixed shapes
• Equipartition of several measures
• Lifting methods in mass partition problems
• Slicing convex sets and measures by a hyperplane