  Affine invariant points and maps for sets were introduced by Gr\"unbaum to
study the symmetry structure of convex sets. We extend these notions to a
functional setting. The role of symmetry of the set is now taken by evenness of
the function. We show that among the examples for affine invariant points are
the classical center of gravity of a log-concave function and its Santal\'o
point. We also show that the recently introduced floating functions and the
John- and L\"owner functions are examples of affine invariant maps. Their
centers provide new examples of affine invariant points for log-concave
functions.
