  The classic Riesz representation theorem characterizes all linear and
increasing functionals on the space $C_{c}(X)$ of continuous compactly
supported functions. A geometric version of this result, which characterizes
all linear increasing functionals on the set of convex bodies in
$\mathbb{R}^{n}$, was essentially known to Alexandrov. This was used by
Alexandrov to prove the existence of mixed area measures in convex geometry.
  In this paper we characterize linear and increasing functionals on the class
of log-concave functions on $\mathbb{R}^{n}$. Here "linear" means linear with
respect to the natural addition on log-concave functions which is the
sup-convolution. Equivalently, we characterize pointwise-linear and increasing
functionals on the class of convex functions. For some choices of the exact
class of functions we prove that there are no non-trivial such functionals. For
another choice we obtain the expected analogue of the result for convex bodies.
And most interestingly, for yet another choice we find a new unexpected family
of such functionals.
  Finally, we explain the connection between our results and recent work done
in convex geometry regarding the surface area measure of a log-concave
functions. An application of our results in this direction is also given.
