Functional equations on convex sets (Q1897515)

Let \(X\) be a real or complex linear space and \(K \subset X\) be a convex absorbing balanced subset. The authors deal with the question under which conditions a function \(f : K \to \mathbb{C}\) has the property that for given \(\alpha \in ]0,1[\), \(\alpha \neq 1/2\), the Jensen difference \[ f \bigl( \alpha x + (1 - \alpha) y \bigr) - \alpha f(x) - (1 - \alpha) f(y) \] is symmetric in \(x\) and \(y\). In characterizing those functions they also deal with the structure of ``local'' polynomials, i.e. solutions \(g\) of the equation \[ \Delta_y^{n + 1} g(x) = 0 \] on certain subsets of \(X\).