  Let $\Omega \subset \mathbb{R}^2$ be a bounded, convex domain and let
$-\Delta \phi_1 = \mu_1 \phi_1$ 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 $x_1, x_2 \in \Omega$ satisfy $\|x_1 -
x_2\| = \mbox{diam}(\Omega)$, then every maximum and minimum is assumed within
distance $c\cdot \mbox{inrad}(\Omega)$ of $x_1$ and $x_2$, where $c$ is a
universal constant (which is the optimal scaling up to the value of $c$).
