Polytopes, Invariants and Harmonic Functions

. The classical harmonic functions are characterized in terms of the mean value property with respect to the unit ball. Replacing the ball by a polytope, we are led to the notion of polyhedral har monic functions, i.e., those continuous functions which satisfy the mean value property with respect to a given polytope. The study of polyhedral harmonic functions involves not only analysis but also algebra, including combinatorics of polytopes and invariant theory for finite reflection groups. We give a brief survey on this subject, focusing on some recent results obtained by the author.