2021/05/19 by Liran Rotem, Rotem, Liran
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Limits and Structures in Graph Theory #Mathematical Inequalities and Applications #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2105.09168
openalex publication_date 2021/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The classic Riesz representation theorem characterizes all linear and increasing functionals on the space Cc(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 ℝn, 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 ℝn. 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.