Hot Spots in Convex Domains are in the Tips (up to an Inradius)

Let $Ω\subset \mathbb{R}^2$ be a bounded, convex domain and let $-Δφ_1 = μ_1 φ_1$ be the first nontrivial Laplacian eigenfunction with Neumann boundary conditions. The Hot Spots conjecture claims that the maximum and minimum are attained at the boundary. We show that they are attained far away from one another: if $x_1, x_2 \in Ω$ satisfy $\|x_1 - x_2\| = \mbox{diam}(Ω)$, then every maximum and minimum is assumed within distance $c\cdot \mbox{inrad}(Ω)$ of $x_1$ and $x_2$, where $c$ is a universal constant (which is the optimal scaling up to the value of $c$).