Non-convexity of the maximal wrappable volume

Establish that for any connected planar region D in R^2 with positive area and no infinitely thin parts, among all connected compact solids B in R^3 that can be enclosed by a 1-Lipschitz image of D with ∂D glued to itself to form a closed surface, the supremum of Vol(B) is never achieved by a convex body; equivalently, prove that for every convex solid B wrappable by D, there exists a non-convex solid B′, also wrappable by D, with strictly larger volume.

Background

The paper studies the maximal volume of a three-dimensional solid that can be wrapped by a given planar sheet of paper, where the sheet may be folded or crumpled but not stretched or torn. Formally, wrapping is modeled via 1-Lipschitz images of the sheet whose boundary is glued to itself to form a closed surface enclosing the solid. The authors distinguish neat wraps (isometric) from more general non-neat wraps that allow crumpling.

The conjecture asserts that convex bodies are never optimal for maximizing enclosed volume for a fixed sheet; rather, a non-convex body always outperforms any convex candidate for the same sheet. This goes beyond existing results on isometric foldings and inflation of polyhedral surfaces (Bleecker; Pak) and contrasts with a reverse problem resolved for convex targets (Karasev), highlighting a gap in current understanding for the forward maximization problem under non-neat wrapping. The paper also discusses a disk-wrapping sphere subquestion as a potential concrete witness of the conjecture.