A classification of $\operatorname{SL}(n)$ 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 $\operatorname{SL}(n)$ and translation invariant valuations on log-concave functions.
Translation invariant valuations on quasi-concave functions
Volume, Polar Volume and Euler Characteristic for Convex Functions
Extensions of translation invariant valuations on polytopes
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