A generalization of the GGR conjecture

For each positive integer n n , function f f , and point c c , the GGR Theorem states that f f is n n times Peano differentiable at c c if and only if f f is n − 1 n-1 times Peano differentiable at c c and the following n n -th generalized Riemann derivatives of f f at c c exist: \[ lim h → 0 1 h n ∑ i = 0 n ( − 1 ) i ( n i ) f ( c + ( n − i − k ) h ) , \lim _{h\rightarrow 0}\frac 1{h^{n}}\sum _{i=0}^n(-1)^i\binom {n}{i}f(c+(n-i-k)h), \] for k = 0 , … , n − 1 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 n -th Riemann smoothness conditions that can play the role of the above set in the GGR Theorem.