Convex sets can have interior hot spots

TL;DR

This paper constructs counterexamples to the hot spots conjecture in high dimensions, showing that convex sets can have interior points where the first non-trivial Neumann eigenfunction attains its maximum, challenging previous assumptions.

Contribution

It extends the hot spots conjecture to log-concave measures and provides counterexamples in all sufficiently large dimensions.

Findings

Counterexamples exist for large dimensions

The hot spots conjecture does not hold universally in convex domains

Extension of the conjecture to log-concave measures

Abstract

The hot spots conjecture asserts that for any convex bounded domain $\Omega$ in $\mathbb R^d$, the first non-trivial Neumann eigenfunction of the Laplace operator in $\Omega$ attains its maximum at the boundary. We construct counterexamples to the conjecture for all sufficiently large values of $d$. The construction is based on an extension of the conjecture from convex sets to log-concave measures.