Volume Maximizing Convex Shape

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $C$ be a convex planar region of area one, with polygonal or smooth boundary. Choose boundary points $x,y$ whose two intervening boundary arcs have equal length. A perimeter-halving folding identifies these arcs in opposite directions by arclength, gluing the arc from $x$ to $y$ to that from $y$ to $x$. The resulting intrinsic surface has the unique convex realization in $\mathbb R^3$ described in the source, allowing a flat realization of volume zero.

Statement recorded in the supplied source.

Among all such unit-area convex regions $C$ and all their perimeter-halving foldings, determine the shape that gives the largest enclosed three-dimensional volume and the maximum volume. [S2]

Short English statement

Among all such unit-area convex regions $C$ and all their perimeter-halving foldings, determine the shape that gives the largest enclosed three-dimensional volume and the maximum volume.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5500062 (AMR-054-0062). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Erik D. Demaine, Joseph S. B. Mitchell and Joseph O'Rourke (editors), The Open Problems Project, maintained original source, Problem 62, Statement and complete Partial/Related Results. https://github.com/edemaine/topp/blob/main/Problems/P.000062.