  A new class of continuous valuations on the space of convex functions on
$\mathbb{R}^n$ is introduced. On smooth convex functions, they are defined for
$i=0,\dots,n$ by \begin{equation*} u\mapsto \int_{\mathbb{R}^n}
\zeta(u(x),x,\nabla u(x))\,[\operatorname{D}^2 u(x)]_i\,{\rm d} x
\end{equation*} where $\zeta\in
C(\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n)$ and $[\operatorname{D}^2
u]_i$ is the $i$th elementary symmetric function of the eigenvalues of the
Hessian matrix, $\operatorname{D}^2 u$, of $u$. Under suitable assumptions on
$\zeta$, these valuations are shown to be invariant under translations and
rotations on convex and coercive functions.

Metric Geometry;Functional Analysis