The Two Hyperplane Conjecture

TL;DR

This paper proposes the Two Hyperplane Conjecture, suggesting isoperimetric surfaces in convex bodies are bounded by parallel hyperplanes, and explores its implications for the Hots Spots Conjecture and level set properties.

Contribution

It introduces a new conjecture linking isoperimetric surfaces to hyperplanes and relates this to eigenfunction level sets and elliptic variational problems.

Findings

Relates the conjecture to quantitative connectivity of level sets.

Recasts known results in a new framework.

Poses questions rather than providing definitive answers.

Abstract

We introduce a conjecture that we call the {\it Two Hyperplane Conjecture}, saying that an isoperimetric surface that divides a convex body in half by volume is trapped between parallel hyperplanes. The conjecture is motivated by an approach we propose to the {\it Hots Spots Conjecture} of J. Rauch using deformation and Lipschitz bounds for level sets of eigenfunctions. We will relate this approach to quantitative connectivity properties of level sets of solutions to elliptic variational problems, including isoperimetric inequalities, Poincar\'e inequalities, Harnack inequalities, and NTA (non-tangentially accessibility). This paper mostly asks questions rather than answering them, while recasting known results in a new light. Its main theme is that the level sets of least energy solutions to scalar variational problems should be as simple as possible.