A generalization of the GGR conjecture

For each positive integer n, function f, and point c, the GGR Theorem states that f is n times Peano differentiable at c if and only if f is n-1 times Peano differentiable at c and the following n-th generalized Riemann derivatives of f at c exist: \begin{equation*} \lim _{h\rightarrow 0}\frac{1}{h^{n}}\sum _{i=0}^n(-1)^i\binom{n}{i}f(c+(n-i-k)h), \end{equation*} for k=0,\ldots ,n-1. The theorem has been recently proved by Ash and Catoiu [Int. Math. Res. Not. IMRN 2022, no. 10, pp. 7893–7921] and has been a conjecture by Ginchev, Guerraggio, and Rocca [Equivalence of Peano and Riemann derivatives, Generalized convexity and optimization for economic and financial decisions (Verona, 1998), Pitagora, Bologna, 1999] since 1998. We provide a new proof of this theorem, based on a generalization of it that produces numerous new sets of n-th Riemann smoothness conditions that can play the role of the above set in the GGR Theorem.