A Doubly Exponentially Crumbled Cake

TL;DR

This paper investigates a cake-cutting game where Alice places points and Bob cuts rectangles, proving that preventing Bob from gaining a significant portion of the cake requires Alice to use an extremely large number of points, growing doubly exponentially.

Contribution

It establishes a lower bound on the number of points Alice needs to prevent Bob from securing a positive fraction of the cake, showing doubly exponential growth.

Findings

Preventing Bob from gaining a fraction 1/r requires at least 2^{2^{Ω(r)}} points.

The conjecture that Bob can always secure at least half the cake remains unresolved.

The results highlight the complexity of controlling Bob's share through point placement.

Abstract

We consider the following cake cutting game: Alice chooses a set P of n points in the square (cake) [0,1]^2, where (0,0) is in P; Bob cuts out n axis-parallel rectangles with disjoint interiors, each of them having a point of P as the lower left corner; Alice keeps the rest. It has been conjectured that Bob can always secure at least half of the cake. This remains unsettled, and it is not even known whether Bob can get any positive fraction independent of n. We prove that if Alice can force Bob's share to tend to zero, then she must use very many points; namely, to prevent Bob from gaining more than 1/r of the cake, she needs at least 2^{2^{\Omega(r)}} points.