Research Problems in Function Theory — Problem 7.57

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.

Fix $n\ge2$. Put $H_\pm=\{x\in\mathbb R^n:\pm x_n>0\}$ and $\sigma(x_1,\ldots,x_n)=(x_1,\ldots,x_{n-1},-x_n)$. For nonempty compact $E\subset H_+$ and $F\subset H_-$ set $F^*=\sigma(F)$. For sets $A,B$, let $\Delta(A,B)$ be all curves in the one-point compactification $\mathbb R^n\cup\{\infty\}$ joining them. The $n$-modulus is \[M_n(\Delta)=\inf_\rho\int_{\mathbb R^n}\rho(x)^n\,dx, \] where $\rho:\mathbb R^n\to[0,\infty]$ is Borel and $\int_\gamma\rho\,ds\ge1$ for every locally rectifiable joining curve; nonrectifiable curves impose no condition. The infimum is $+\infty$ when there is no admissible density.

Statement recorded in the supplied source.

Does \[M_n(\Delta(E,F))\le M_n(\Delta(E,F^*)) \] hold for every such pair of compact sets? [S2]

Short English statement

Does reflecting the lower of two compact sets across the separating hyperplane increase the modulus of the family of curves joining the sets?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307057 (AMR-022-7057). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), Problem 7.57, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.