# Geometric properties of optimizers for the maximum gradient of the torsion function

arXiv: https://arxiv.org/abs/2512.09400 · HTML: https://mlsanity.com/paper/2512.09400/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Dec 2025.

## Built on by (most cited first, 1 of 1)

- [Parametrizing Convex Sets Using Sublinear Neural Networks](https://mlsanity.com/paper/2605.03520/) (2026)

## Abstract (arXiv)

Consider J(Omega):= |nabla u_Omega|_∞/√|Omega|  and JP(Omega):= |nabla u_Omega|_∞/P(Omega) , where Omega is a planar convex domain, u_Omega is the torsion function, P(Omega) is the perimeter of Omega and |Omega| its area. We prove that there exist planar convex domains that maximize the functionals J and JP, and any maximizer has a C1 boundary that contains a line segment on which |nabla u_Omega| attains its maximum.