Impossibility of a nontrivial Brunn--Minkowski inequality for higher Dirichlet eigenvalue

Let ${\lambda }_j(K)$ be the $j$th Dirichlet eigenvalue of a convex body $K$. It is well known that ${\lambda }_1$ satisfies a Brunn–Minkowski inequality: $K\mapsto {\lambda }_1(K{)}^{-1\mathrm{/}2}$ is concave on the family of convex bodies. We show that no analogous statement holds for higher eigenvalues. More precisely, for any $j\ge 2$ and $N\ge 2$, if $K\mapsto (f\circ {\lambda }_j)(K)$ is concave on the family of convex bodies in ${\mathbb{R}}^N$ for some function $f:(0,\mathrm{\infty })\to \mathbb{R}$, then $f$ must be constant.