Mahler's Conjecture

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.

A convex body $K\subset\mathbb R^n$ is a compact convex set with nonempty interior. In the symmetric formulation, its center is moved to the origin, so $K=-K$. Its polar body is \[ K^\circ=\{y\in\mathbb R^n:\langle x,y\rangle\le1 \text{ for every }x\in K\}, \] where $\langle\cdot,\cdot\rangle$ is the Euclidean inner product. Write $|K|$ for its $n$-dimensional volume. The cube $[-1,1]^n$ and its polar, the cross-polytope $\{x:\sum_i|x_i|\le1\}$, have volume product $4^n/n!$.

Statement recorded in the supplied source.

For every $n\ge1$ and every origin-symmetric convex body $K\subset\mathbb R^n$, must \[ |K|\,|K^\circ|\ge\frac{4^n}{n!}? \] Equivalently, do the cube and the cross-polytope attain the smallest possible volume product in the centrally symmetric class? [S2]

Short English statement

Does every centrally symmetric convex body have volume product at least that of a cube, equivalently that of its polar cross-polytope?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1122 (GEO-002). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hiroshi Iriyeh and Masataka Shibata, Symmetric Mahler's conjecture for the volume product in the three dimensional case, arXiv:1706.01749, revised 2019, Section 1.1, displayed symmetric conjecture (2). https://arxiv.org/abs/1706.01749.