Research Problems in Function Theory — Problem 4.23

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 $m\ge2$, $n\ge1$, and let $x_1,\ldots,x_n\in\mathbb R^m$ be arbitrary points, allowing coincidences. With $|\cdot|$ the Euclidean norm, put \[E(x_1,\ldots,x_n)=\left\{x\in\mathbb R^m:\prod_{i=1}^n|x-x_i|\le1\right\},\qquad V_{m,n}=\sup_{x_1,\ldots,x_n}\lambda_m(E(x_1,\ldots,x_n)), \] where $\lambda_m$ is $m$-dimensional Lebesgue measure.

Statement recorded in the supplied source.

Determine $V_{m,n}$ and the point configurations attaining the maximum. Separately, if $E(x_1,\ldots,x_n)$ is connected, must there be a centre $a\in\mathbb R^m$ for which \[E(x_1,\ldots,x_n)\subseteq\{x:|x-a|\le2\}?\] Both the volume and connected-set covering questions are included; the source records the covering conclusion in dimension two. [S2]

Short English statement

Maximize the volume of a product-of-distances sublevel set in higher dimensions and decide whether every connected such set fits inside a ball of radius two.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2304023 (AMR-022-4023). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory, arXiv:1809.07200, 2018. Problem 4.23, its complete Update 4.23, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.