Valuations on Log-Concave Functions

A classification of $\mathrm{SL}(n)$ and translation covariant Minkowski valuations on log-concave functions is established. The moment vector and the recently introduced level set body of log-concave functions are characterized. Furthermore, analogs of the Euler characteristic and volume are characterized as $\mathrm{SL}(n)$ and translation invariant valuations on log-concave functions.