How to cut a cake with a gram matrix

TL;DR

This paper explores fair division using a gram matrix approach, providing explicit bounds, a solution to an open problem, and methods for constructing fair divisions with piecewise constant measures.

Contribution

It offers a new proof of a generalized fair division concept, explicit bounds, and an effective construction method for piecewise constant measures.

Findings

Provided a new proof of Barbanel's generalized super envy-free division

Derived explicit bounds for fair division

Developed a method to construct fair divisions with piecewise constant densities

Abstract

In this article we study the problem of fair division. In particular we study a notion introduced by J. Barbanel that generalizes super envy-free fair division. We give a new proof of his result. Our approach allows us to give an explicit bound for this kind of fair division. Furthermore, we also give a theoretical answer to an open problem posed by Barbanel in 1996. Roughly speaking, this question is: how can we decide if there exists a fair division satisfying some inequalities constraints? Furthermore, when all the measures are given with piecewise constant density functions then we show how to construct effectively such a fair division.