Convex bodies: the brunn–minkowski theory

In this paper, we give a short survey on discrete geometry on red and blue points in the plane, most of whose results were obtained in the past decade. We consider balanced subdivision problems, geometric graph problems, graph embedding problems, Gallai-type problems and others.

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Discrete Geometry on Red and Blue Points in the Plane — A Survey —

We describe a regular cell complex model for the configuration space F(ℝ
 d
 , n). Based on this, we use Equivariant Obstruction Theory to prove the prime power case of the conjecture by Nandakumar and Ramana Rao that every polygon can be partitioned into n convex parts of equal area and perimeter.

Convex equipartitions via Equivariant Obstruction Theory

This paper is part of an emerging line of work at the intersection of machine learning and mechanism design, which aims to avoid noise in training data by correctly aligning the incentives of data sources. Specifically, we focus on the ubiquitous problem of linear regression, where strategyproof mechanisms have previously been identified in two dimensions. In our setting, agents have single-peaked preferences and can manipulate only their response variables. Our main contribution is the discovery of a family of group strategyproof linear regression mechanisms in any number of dimensions, which we call generalized resistant hyperplane mechanisms. The game-theoretic properties of these mechanisms --- and, in fact, their very existence --- are established through a connection to a discrete version of the Ham Sandwich Theorem.

https://dl.acm.org/doi/pdf/10.1145/3219166.3219175

Strategyproof Linear Regression in High Dimensions

We show that, for any prime power $n$ and any convex body $K$ (i.e., a compact convex set with interior) in $\mathbb{R }^d$, there exists a partition of $K$ into $n$ convex sets with equal volumes and equal surface areas. Similar results regarding equipartitions with respect to continuous functionals and absolutely continuous measures on convex bodies are also proven. These include a generalization of the ham-sandwich theorem to arbitrary number of convex pieces confirming a conjecture of Kaneko and Kano, a similar generalization of perfect partitions of a cake and its icing, and a generalization of the Gromov–Borsuk–Ulam theorem for convex sets in the model spaces of constant curvature.

Convex equipartitions: the spicy chicken theorem

Given convex bodies K1,…,Kd in ℝd and numbers α1,…,αd∈[0,1], we give a sufficient condition for existence and uniqueness of an (oriented) halfspace H with Vol (H∩Ki)=αi⋅Vol Ki for every i. The result is extended from convex bodies to measures.

/pdf/slicing-convex-sets-and-measures-by-a-hyperplane-bz0do8wcuh.pdf

Slicing Convex Sets and Measures by a Hyperplane

Following ideas of Iriyeh and Shibata we give a short proof of the three-dimensional Mahler conjecture for symmetric convex bodies. Our contributions include, in particular, simple self-contained proofs of their two key statements. The first of these is an equipartition (ham sandwich type) theorem which refines a celebrated result of Hadwiger and, as usual, can be proved using ideas from equivariant topology. The second is an inequality relating the product volume to areas of certain sections and their duals. Finally we give an alternative proof of the characterization of convex bodies that achieve the equality case and establish a new stability result.

Equipartitions and Mahler Volumes of Symmetric Convex Bodies

We generalize the ham sandwich theorem for the case of well separated measures. Given convex bodies $K_1,...,K_d$ in $\mathbb{R_d}$ and numbers $\alpha_1,...,\alpha_d \in [0, 1]$, we give a sufficient condition for existence and uniqueness of an (oriented)halfspace H with Vol($H \cap K_i$)= $\alpha_i \dot$ Vol$K_i$ for every i. The result is extended from convex bodies to measures.

Slicing convex sets and measures by a hyperplane

We show that, for any prime power p^k and any convex body K (i.e., a compact convex set with interior) in Rd, there exists a partition of K into p^k convex sets with equal volume and equal surface area. We derive this result from a more general one for absolutely continuous measures and continuous functionals on the space of convex bodies. This result was independently found by Roman Karasev generalizing work of Gromov and Memarian who proved it for the standard measure on the sphere and p=2. Our proof uses basics from the theory of optimal transport and equivariant topology. The topological ingredient is a Borsuk-Ulam type statement on configuration space with the standard action of the symmetric group. This result was discovered in increasing generality by Fuks, Vaseliev and Karasev. We include a detailed proof and discuss how it relates to Gromov's proof for the case p=2.

Convex Equipartitions of volume and surface area

We show that when $n$ is a power of two, any $nD$ measures in $\mathbb{R}^n$ can be bisected by an arrangement of $D$ hyperplanes.

Alfredo Hubard

Papers

Slicing Convex Sets and Measures by a Hyperplane

Equipartitions and Mahler Volumes of Symmetric Convex Bodies

Slicing convex sets and measures by a hyperplane

Convex Equipartitions of volume and surface area

Bisecting measures with hyperplane arrangements