Reverse Banaszczyk minimization and the ellipsoidal regime

Definitions and mathematical statement

Fix a dimension $d\geq1$ and a positive length bound $c$. For a Borel set $K\subseteq\mathbb R^d$, its standard Gaussian volume is \[ \gamma_d(K)=(2\pi)^{-d/2}\int_K e^{-\|x\|_2^2/2}\,dx. \] A convex body here means a compact convex set with nonempty interior. Say that $K$ has property $B_c$ if for every integer $m\geq0$ and every sequence $v_1,\ldots,v_m\in\mathbb R^d$ with $\|v_i\|_2\leq c$, there are signs $\epsilon_1,\ldots,\epsilon_m\in\{-1,1\}$ for which \[ \sum_{i=1}^m\epsilon_i v_i\in K. \] For $m=0$ the sum is $0$, so the property includes $0\in K$. Sequences may repeat vectors, and the signing may depend on the entire sequence.

Determine the extremal quantity \[ b_d(c)=\inf\{\gamma_d(K):K\text{ is a convex body with }B_c\}, \] and identify bodies attaining, or approaching, this infimum. The source treats $c$ as a fixed small absolute constant; displaying it makes the dependence on the vector normalization explicit. One may then ask how the smallest volume behaves with dimension for that fixed choice of $c$. The universal requirement concerns every sequence length, not just a prescribed number of vectors. No central symmetry is imposed unless one explicitly studies that additional restricted variant; a signed-sum requirement alone does not define symmetry of the body.

Short English statement

How small can the Gaussian volume of a convex body be if it can receive a signed sum of every bounded sequence of vectors?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000921 (AIM-COMBINATORICS-0046), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Hereditary discrepancy and factorization norms, item 1.8. Original workshop question underlying AIM-COMBINATORICS-0046. http://aimpl.org/hereddiscrep/1/.