Let α1,...,αm be two or more positive reals with sum 1, let C ⊆ Rk be an open convex set, and f : C → Rk be a continuous injection with convex image. For each nonempty set S ⊆ C, let M(S) be the family of quasi-arithmetic means of all m-tuples of vectors in C with respect to f and the weights α1,...,αm, that is, the family M(S) = f −1 (α1f(x1) + ··· + αmf(xm)) : x1,...,xm ∈ S. We provide a simple necessary and sufficient condition on S for which the infinite iterationn n Mn(S) is relatively dense in the convex hull of S.
Quasi-arithmetic means ad libitum
Paolo Leonetti
2023-01-01
Abstract
Let α1,...,αm be two or more positive reals with sum 1, let C ⊆ Rk be an open convex set, and f : C → Rk be a continuous injection with convex image. For each nonempty set S ⊆ C, let M(S) be the family of quasi-arithmetic means of all m-tuples of vectors in C with respect to f and the weights α1,...,αm, that is, the family M(S) = f −1 (α1f(x1) + ··· + αmf(xm)) : x1,...,xm ∈ S. We provide a simple necessary and sufficient condition on S for which the infinite iterationn n Mn(S) is relatively dense in the convex hull of S.File | Dimensione | Formato | |
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