On the maximum volume solid wrappable by a given sheet of paper

TL;DR

This paper investigates the maximum volume of 3D solids that can be wrapped with a given 2D sheet, proposing a conjecture that non-convex bodies can achieve larger volumes than convex ones.

Contribution

It introduces a conjecture suggesting non-convex solids can always surpass convex ones in maximum volume when wrapped by a given sheet.

Findings

Conjecture that non-convex bodies can have greater volume than convex bodies wrapped by the same sheet.

Discussion of a key subproblem involving the sphere.

Exploration of related work and future research directions.

Abstract

We consider the problem of wrapping three-dimensional solid bodies with a given planar sheet of paper, where the paper may be folded or wrinkled but not stretched or torn. We propose a conjecture characterising the maximumvolume solid wrappable by any given sheet: the maximum is always achieved (or approached) by a non-convex body. In other words, for any convex solid wrappable by a given sheet, there exists a non-convex solid of strictly greater volume that the same sheet can wrap. We discuss related work, a key subquestion involving the sphere, and several further directions.