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.
2023
2023
https://link.springer.com/article/10.1007/s00025-023-02009-y
Mathematics Subject Classification: 11B05, 26B25 52B55, 03D25. Quasi-arithmetic mean, Convexity, Quasi-convexity, Minkowsky sum.
Leonetti, Paolo
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2163275
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