  Let $D \subset \mathbb{R}^d$ be a bounded, connected domain with smooth
boundary and let $-\Delta u = \mu_1 u$ be the first nontrivial eigenfunction of
the Laplace operator with Neumann boundary conditions. We prove $$ \max_{x \in
D} ~u(x) \leq 60 \cdot \max_{x \in \partial D} ~u(x)$$ and emphasize that this
constant is uniform among all connected domains with smooth boundary in all
dimensions. In particular, the Hot Spots Conjecture cannot fail by an arbitrary
factor. The inequality also holds for other (Neumann-)eigenfunctions (possibly
with a different constant) provided the eigenvalue is smaller than the first
Dirichlet eigenvalue. An example of Kleefeld shows that the optimal constant is
at least $1 + 10^{-3}$.
