Convex sets can have interior hot spots

The hot spots conjecture asserts that for any convex bounded domain $\mathrm{\Omega }$ in ${\mathbb{R}}^d$, the first non-trivial Neumann eigenfunction of the Laplace operator in $\mathrm{\Omega }$ 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.