Projections of log-concave functions

Recently, it has been proven in [V. Milman, A. Segal and B. Slomka, A characterization of duality through section/projection correspondence in the finite dimensional setting, J. Funct. Anal. 261(11) (2011) 33663389] that the well-known duality mapping on the class of closed convex sets in ⁿ containing the origin is the only operation, up to obvious linear modifications, that interchanges linear sections with projections. In this paper, we extend this result to the class of geometric log-concave functions (attaining 1 at the origin). As the notions of polarity and the support function were recently uniquely extended to this class by Artstein-Avidan and Milman, a natural notion of projection arises. This notion of projection is justified by our result. As a consequence of our main result, we prove that, on the class of lower semi continuous non-negative convex functions attaining 0 at the origin, the polarity operation is the only operation interchanging addition with geometric inf-convolution and the support function is the only operation interchanging addition with inf-convolution.