Lifting Methods in Mass Partition Problems

@article{Soberon2022LiftingMI,
  title={Lifting Methods in Mass Partition Problems},
  author={Pablo Sober'on and Yuki Takahashi},
  journal={International Mathematics Research Notices},
  year={2022}
}
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… 
1 Citations

Figures from this paper

Bisections of mass assignments using flags of affine spaces

We use recent extensions of the Borsuk–Ulam theorem for Stiefel manifolds to generalize the ham sandwich theorem to mass assignments. A k-dimensional mass assignment continuously imposes a measure on

References

SHOWING 1-10 OF 50 REFERENCES

Convex equipartitions: the spicy chicken theorem

We show that, for any prime power $$n$$n and any convex body $$K$$K (i.e., a compact convex set with interior) in $$\mathbb{R }^d$$Rd, there exists a partition of $$K$$K into $$n$$n convex sets with

Equipartitions with Wedges and Cones

TLDR
A Borsuk-Ulam theorem for flag manifolds is introduced, which the author believes to be of independent interest, and a stronger statement, namely that any $d$ mass distributions in $\mathbb{R}^d$ can be simultaneously bisected by a single hyperplane is proved.

Borsuk-Ulam theorems for products of spheres and Stiefel manifolds revisited

We give a different and possibly more accessible proof of a general Borsuk--Ulam theorem for a product of spheres, originally due to Ramos. That is, we show the non-existence of certain

More Bisections by Hyperplane Arrangements

TLDR
A different proof of the Hubard and Karasev result is given using the framework of Blagojević, Frick, Haase & Ziegler (2016), based on the equivariant relative obstruction theory of tom Dieck, which was developed for handling the Grünbaum–Hadwiger–Ramos hyperplane measure partition problem.

BALANCED CONVEX PARTITIONS OF MEASURES IN ℝ d

We prove the following generalization of the ham sandwich theorem, conjectured by Imre Barany. Given a positive integer k and d nice measures μ 1 , μ 2 ,…, μ d in ℝ d such that μ i (ℝ d )= k for all

Appollonius Revisited: Supporting Spheres for Sundered Systems

TLDR
If there exists a unique Euclidean sphere that is simultaneously a near support for each member of ${\cal B}'$ and a far support for the selected points are affinely independent and hence form the vertex-set of a d-simplex.

Generalizations of the Yao-Yao partition theorem and the central transversal theorem

We generalize the Yao–Yao partition theorem by showing that for any smooth measure in Rd there exist equipartitions using (t+ 1)2d−1 convex regions such that every hyperplane misses the interior of

Bisection of Circle Colorings

Consider $2n$ beads of k colors arranged on a necklace, using $2a$, beads of color i. A bisection is a set of disjoint strings (“intervals”) of beads whose union captures half the beads of each

Measure partitions using hyperplanes with fixed directions

We study nested partitions of Rd obtained by successive cuts using hyperplanes with fixed directions. We establish the number of measures that can be split evenly simultaneously by taking a partition